# Global regularity in the Monge-Ampère obstacle problem

Shibing Chen  Shibing Chen, School of Mathematical Sciences, University of
Science and Technology of China, Hefei, 230026, P.R. China.
<EMAIL_ADDRESS>,  Jiakun Liu  Jiakun Liu, School of Mathematics and
Applied Statistics, University of Wollongong, Wollongong, NSW 2522, AUSTRALIA
<EMAIL_ADDRESS>and  Xianduo Wang  Xianduo Wang, Department of Mathematical
Sciences, Tsinghua University, Beijing 100084, China  xd-
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we establish the global $W^{2,p}$ estimate for the Monge-Ampère
obstacle problem: $(Du)_{\sharp}f\chi{{}_{\\{u>\frac{1}{2}|x|^{2}\\}}}=g$,
where $f$ and $g$ are positive continuous functions supported in disjoint
bounded $C^{2}$ uniformly convex domains $\overline{\Omega}$ and
$\overline{\Omega^{*}}$, respectively. Furthermore, we assume that
$\int_{\Omega}f\geq\int_{\Omega^{*}}g$. The main result shows that
$Du:\overline{U}\rightarrow\overline{\Omega^{*}}$, where
$U=\\{u>\frac{1}{2}|x|^{2}\\}$, is a $W^{1,p}$ diffeomorphism for any
$p\in(1,\infty)$. Previously, it was only known to be a continuous
homeomorphism according to Caffarelli and McCann [5]. It is worth noting that
our result is sharp, as we can construct examples showing that even with the
additional assumption of smooth densities, the optimal map $Du$ is not
Lipschitz. This obstacle problem arises naturally in optimal partial
transportation.

###### Key words and phrases:

Optimal transportation, Monge-Ampère equation, free boundary

###### 2000 Mathematics Subject Classification:

35J96, 35J25, 35B65.

Research of Chen was supported by National Key R&D program of China
2022YFA1005400, 2020YFA0713100, National Science Fund for Distinguished Young
Scholars (No. 12225111), and NSFC No. 12141105. Research of Liu was supported
by ARC DP200101084, DP230100499 and FT220100368.

##  1\. introduction

In this paper we study the convex solution to the Monge-Ampère obstacle
problem

(1.1) $(Du)_{\sharp}f\chi_{{}_{\\{u>\frac{1}{2}|x|^{2}\\}}}=g,$

where $0\leq f,g\in L^{1}(\mathbb{R}^{n})$ are supported in the closures of
bounded convex domains $\Omega,\Omega^{*}$, respectively. We also assume that
$\overline{\Omega}$ and $\overline{\Omega^{*}}$ are disjoint, and that
$\int_{\Omega}f\geq\int_{\Omega^{*}}g.$ The equation (1.1) is understood in
Brenier’s sense, namely, a convex function $u$ is a solution of (1.1) if and
only if for each bounded continuous function
$\varphi:\mathbb{R}^{n}\rightarrow\mathbb{R},$ we have

$\int_{{}_{\\{u>\frac{1}{2}|x|^{2}\\}}}\varphi(Du(x))f(x)dx=\int_{\mathbb{R}^{n}}\varphi(y)g(y)dy.$

This obstacle problem arises in the optimal partial transport problem. A non-
negative, finite Borel measure $\gamma$ on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ is called a transport plan (with mass
$m:=\int_{\Omega^{*}}g$) from the distribution $(\Omega,f)$ to the
distribution $(\Omega^{*},g)$, if
$\gamma(\mathbb{R}^{n}\times\mathbb{R}^{n})=m$ and

(1.2)
${\begin{split}\gamma(A\times\mathbb{R}^{n})\leq\int_{A\cap\Omega}f(x)\,dx,\ \
\ \gamma(\mathbb{R}^{n}\times
A)\leq\int_{A\cap\Omega^{*}}g(y)\,dy\end{split}}$

for any Borel set $A\subset\mathbb{R}^{n}$. A transport plan $\gamma$ is
optimal if it minimises the cost functional

(1.3) $\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}|x-y|^{2}\,d\gamma(x,y)$

among all transport plans. In [5], Caffarelli and McCann have shown the
existence and uniqueness of solutions to both (1.1) and (1.3), moreover, the
optimal plan of (1.3) can be characterised as follows

(1.4) $\gamma=(\text{Id}\times Du)_{\\#}f\chi_{U},$

where $u$ is a convex solution to (1.1) and
$U:=\\{u>\frac{1}{2}|x|^{2}\\}\cap\Omega.$

In [5], the authors have established the interior $C^{1,\alpha}$ regularity of
free boundary $\mathcal{F}:=\partial\\{u>\frac{1}{2}|x|^{2}\\}\cap\Omega$,
assuming the domains are strictly convex and densities are bounded from below
and above. Then, in [9], the authors proved the interior higher order
regularity of free boundary assuming the domains are smooth uniformly convex
and densities are positive smooth. For the global regularity of $\mathcal{F}$
or $u,$ it was proved in [5] that the free boundary is $C^{1}$ regular up to
its intersection with the fixed boundary, and that $Du$ is a $C^{0}$
homeomorphism between $\overline{U}$ and $\overline{\Omega^{*}}.$ However,
nothing more is known concerning the finer regularity of free boundary up to
its intersection with the fixed boundary and the global regularity of the
solution $u$ in $U.$ There are many basic questions left open, for instance it
is even not known whether the active region is a Lipschitz domain.

In this work, we investigate the above mentioned questions systematically. As
an initial step, we first prove that the free boundary always intersects the
fixed boundary in a nice way. Indeed, it is mentioned in [5, below (7.2)] that
it is not clear that whether the nontransverse intersection points between the
free boundary and the fixed boundary exist or not. In [5], the set of
nontransverse intersection points is specified as

(1.5)
$\partial_{nt}\Omega:=\\{x\in\partial\Omega\cap\overline{\Omega\cap\partial
U}:\langle Du(x)-x,z\rangle\leq 0\ \ \forall z\in\Omega\\}.$

Note that in dimension two, the active domain $U$ looks like a cusp at such a
nontransverse intersection point. It is also pointed out in [5] that it is not
clear whether the free boundary is $C^{1,\alpha^{\prime}}$ up to such points.
This issue was later studied by Indrei [16], in which he proved
$\partial_{nt}\Omega$ is of Hausdorff dimension at most $n-2,$ assuming
$\Omega,\Omega^{*}\in C^{1,1}$ and uniformly convex. In this paper we resolve
this question as follows.

######  Theorem 1.1.

Let $\Omega,\Omega^{*}\subset\mathbb{R}^{n}$ be two bounded, strictly convex
domains, with $\overline{\Omega}\cap\overline{\Omega^{*}}=\emptyset$. Assume
that $\lambda^{-1}<f,g<\lambda$ in $\Omega,\Omega^{*}$ respectively for some
positive constant $\lambda.$ Let $u$ be a convex solution to (1.1). Then,
$\partial_{nt}\Omega=\emptyset$, in particular, $\partial U$ is globally
Lipschitz.

A direct consequence of Theorem 1.1 is the $C^{1,\alpha}$ regularity of
$\mathcal{F}$ up to its intersection with the fixed boundary.

######  Corollary 1.1.

Under the same conditions as in Theorem 1.1, we have that $u\in
C^{1,\alpha}(\overline{U})$ and $\overline{\mathcal{F}}$ is $C^{1,\alpha}$
regular for some $\alpha\in(0,1).$

######  Remark 1.1.

_If $\Omega$ and $\Omega^{*}$ partly overlap, namely if
$\Omega\cap\Omega^{*}\neq\emptyset$, Figalli [12, 13] proved that
$\mathcal{F}$ and $\mathcal{F}^{*}$ are locally $C^{1}$ smooth away from the
common region $\Omega\cap\Omega^{*}$. Later, Indrei [16] improved the $C^{1}$
regularity to $C^{1,\alpha^{\prime}}$, also away from $\Omega\cap\Omega^{*}$.
_

To study the higher regularity of $u$ and $\mathcal{F},$ we further assume
that the domains $\Omega,\Omega^{*}$ are $C^{2}$ and uniformly convex, and the
densities $f\in C^{0}(\overline{\Omega}),g\in C^{0}(\overline{\Omega^{*}}).$
For a given point $x_{0}\in\partial U,$ denote
$y_{0}:=Du(x_{0})\in\partial\Omega^{*}.$ If
$x_{0}\in\partial\Omega\setminus\overline{\mathcal{F}},$ then since $\partial
U,\partial\Omega^{*}$ are $C^{2}$ uniformly convex near $x_{0},y_{0}$
respectively, we can apply the argument in [7] to show that $u$ is
$C^{1,1-\epsilon}$ regular near $x_{0}$ for $\epsilon>0$ as small as we want.
If $x_{0}\in\mathcal{F},$ we can apply the argument in [9] to show the
corresponding estimate. If $x_{0}\in\overline{\mathcal{F}}\cap\partial\Omega,$
then $U$ is only Lipschitz at $x_{0},$ moreover $\partial U$ contains fixed
boundary part (convex) and free boundary part (non-convex). Low regularity and
nonconvexity of $\partial U$ near $x_{0}$ bring serious difficulty to prove
the $C^{1,1-\epsilon}$ regularity of $u$. In this paper, we overcome this
difficulty by developing a very delicate blow-up argument at such points, and
finally we establish the following theorem.

######  Theorem 1.2.

Besides the assumptions in Theorem 1.1, assume further that
$\Omega,\Omega^{*}$ are $C^{2}$ and uniformly convex, and that the densities
$f\in C^{0}(\overline{\Omega}),g\in C^{0}(\overline{\Omega^{*}}).$ Then $u\in
W^{2,p}(\overline{U})$ for any $p>1.$ Moreover, the free boundary
$\mathcal{F}$ is $C^{1,1-\epsilon}$ up to its intersection with the fixed
boundary for any $\epsilon\in(0,1).$

######  Remark 1.2.

_In the last section we will provide some example to show that the $W^{2,p}$
regularity of $u$ in $\overline{U}$ is optimal in some sense. The ideas
introduced in this paper can also be used to study the optimal transport
problem when the target consists of two separated convex domains, which will
be done in a forthcoming paper. _

The rest of paper is organised as follows. In Section 2 we recall some results
from [4, 5, 7] which will be used in subsequent sections. In Section 3 we
prove the Lipschitz regularity of the active region $U.$ Section 4, we
establish the obliqueness estimate for points nearby the intersection of the
free boundary and fixed boundary. In section 5 and 6, we prove the global
$C^{1,1-\epsilon}$ estimate of $u$ and then establish the global $W^{2,p}$
estimate of $u$ in $U.$ In the last section, we construct some example showing
that our result is sharp.

##  2\. Preliminaries

###  2.1. Potential functions

Throughout the paper, we always assume that the densities $f,g$ satisfy
$\lambda^{-1}<f,g<\lambda$ in $\Omega,\Omega^{*}$, respectively, for a
positive constant $\lambda$, and the domains
$\Omega,\Omega^{*}\subset\mathbb{R}^{n}$ are bounded, strictly convex and
their closures are disjoint. The source is assumed to contain more mass than
the target, namely, $\int_{\Omega}f\geq\int_{\Omega^{*}}g.$ The active region
is denoted by $U:=\left\\{u>\frac{1}{2}|x|^{2}\right\\}\cap\Omega.$

Let $u:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a convex solution to (1.1).
Replacing $u$ by $\max\\{u,\frac{|x|^{2}}{2}\\},$ we may assume that
$u=\frac{|x|^{2}}{2}$ in $\Omega\setminus\overline{U}.$ Let

$v(y):=\sup_{x\in\Omega}\left\\{y\cdot x-u(x)\right\\}\ \ \text{for}\
y\in\mathbb{R}^{n}.$

Then,

(2.1) $(Dv)_{\\#}(g+f\chi_{{}_{\Omega\backslash\overline{U}}}))=f.$

We may also extend $u$ from $U$ to $\mathbb{R}^{n}$ in the following way

(2.2) $\bar{u}(x):=\sup\\{L(x):L\ \text{is affine},\ L\leq u\ \text{in}\
U,\text{and}\ DL\in\Omega^{*}\\}.$

By (2.1), (2.2) and since $\Omega,\Omega^{*}$ are bounded and convex, $u,v$
are globally Lipschitz in $\mathbb{R}^{n}$ and satisfy

(2.3) $C^{-1}(\chi_{{}_{\Omega\setminus
U}}+\chi_{{}_{\Omega^{*}}})\leq\mbox{det}\,D^{2}v\leq
C(\chi_{{}_{\Omega\setminus U}}+\chi_{{}_{\Omega^{*}}})$

and

(2.4) $C^{-1}\chi_{{}_{U}}\leq\mbox{det}\,D^{2}\bar{u}\leq C\chi_{{}_{U}}$

in the sense of Alexandrov [2], where $C$ is a positive constant depending
only on $\lambda$.

For a convex function $w:\mathbb{R}^{n}\to(-\infty,\infty]$, the associated
_Monge-Ampère measure_ $\mu_{w}$ is defined by

(2.5) $\mu_{w}(E):=\left|\partial w(E)\right|$

for any measurable set $E\subset\mathbb{R}^{n}$, where $\partial w$ is the
sub-gradient of $w$ and $|\cdot|$ denotes the $n$-dimensional Hausdorff
measure. We say that $w$ satisfies
$C_{1}\chi_{{}_{W}}\leq\mbox{det}\,D^{2}w\leq C_{2}\chi_{{}_{W}}$ in the sense
of Alexandrov, if

$C_{1}|E\cap W|\leq\mu_{w}(E)\leq C_{2}|E\cap W|\ \ \ \forall\
E\subset\mathbb{R}^{n}.$

Hence (2.3) implies that the Monge-Ampère measure $\mu_{v}$ is actually
supported and bounded on $({\overline{\Omega}}\setminus
U)\cup\overline{\Omega^{*}}.$

###  2.2. $C^{1,\alpha^{\prime}}$ regularity of $\mathcal{F}$

We recall the _interior ball condition_ proved in [5], which will be useful in
our subsequent analysis.

######  Lemma 2.1 ([5]).

Let $x\in U$ and $y=Du(x)$, then

$\Omega\cap B_{|x-y|}(y)\subset U.$

It is shown in [5] that $u$ is $C^{1}$ smooth up to the free boundary
$\mathcal{F}$, and the unit inner normal vector of $\mathcal{F}$ is given by

(2.6) $\nu(x)=\frac{Du(x)-x}{|Du(x)-x|}\quad\forall\ x\in\mathcal{F}.$

Hence, the regularity of $u$ up to the free boundary $\mathcal{F}$ implies the
regularity of the free boundary $\mathcal{F}$ itself. The following regularity
results have been obtained in [5].

######  Theorem 2.1 ([5]).

Assume that $\Omega,\Omega^{*}$ are disjoint and strictly convex, the
densities $f,g$ satisfy $\lambda^{-1}<f,g<\lambda$ for a positive constant
$\lambda$. Then

  * $i)$

$\bar{u},v\in C^{1}(\mathbb{R}^{n})$, $Dv$ is 1-1 from $\overline{\Omega^{*}}$
to $\overline{U}$, and $Du$ is 1-1 from $\overline{U}$ to
$\overline{\Omega^{*}}$.

  * $ii)$

$u\in C^{1,\alpha^{\prime}}$ up to the free boundary $\mathcal{F}$, and
$\mathcal{F}$ is $C^{1,\alpha^{\prime}}$ for some $\alpha\in(0,1)$.

###  2.3. Sub-level sets

To study higher order regularity of the potentials $u,v$, we introduce the
(centred) sub-level sets as in [3, 4].

######  Definition 2.1.

Let $y_{0}\in\overline{\Omega^{*}}$ and $h>0$ be a small constant. We denote
by

(2.7)
$S^{c}_{h}[v](y_{0}):=\left\\{y\in\mathbb{R}^{n}\,:\,v(y)<v(y_{0})+(y-y_{0})\cdot\bar{p}+h\right\\}$

the centred sub-level set of $v$ with height $h$, where
$\bar{p}\in\mathbb{R}^{n}$ is chosen such that the centre of mass of
$S^{c}_{h}[v](y_{0})$ is $y_{0}$. We denote by

(2.8)
$S_{h}[v](y_{0}):=\left\\{y\in\Omega^{*}\,:\,v(y)<\ell_{y_{0}}(y)+h\right\\}$

the sub-level set of $v$ with height $h$, where $\ell_{y_{0}}$ is a support
function of $v$ at $y_{0}$.

Note that in the above definition, $S_{h}[v](y_{0})$ is a subset of
$\Omega^{*}$ but $S^{c}_{h}[v](y_{0})$ may not be contained in $\Omega^{*}$.
In the following we will write $S_{h}[v](y_{0})$ and $S^{c}_{h}[v](y_{0})$ as
$S_{h}[v]$ and $S^{c}_{h}[v]$ when no confusion arises.

######  Remark 2.1.

_Suppose $v(0)=0,v\geq 0.$ Let $L$ be the affine function such that
$S^{c}_{h}[v](0)=\\{v<L\\}$. Since $(L-v)(0)=h$, $L=v$ on $\partial
S^{c}_{h}[v](0),$ $L\geq v\geq 0$ in $S^{c}_{h}[v](0),$ and $S^{c}_{h}[v](0)$
is balanced around $0$, we have that_

(2.9) $v\leq L\leq Ch\ \ \ \text{in}\ S^{c}_{h}[v](0)$

_for a constant $C$ depending only on $n.$ The same property also holds if $v$
is replaced by $u.$ _

For any $x_{0}\in\mathcal{F}$, we have
$y_{0}:=Du(x_{0})\in\partial\Omega^{*}$. When $h>0$ is sufficiently small, by
[5, Lemma 7.6 and Lemma 7.11] we have

(2.10) $S_{h}^{c}[v](y_{0})\cap\overline{\Omega}=\emptyset.$

By [5, Theorem 7.13] we have furthermore the strict convexity

(2.11) $v(y)\geq
v(y_{0})+Dv(y_{0})\cdot(y-y_{0})+C|y-y_{0}|^{1+\beta}\quad\forall\,y\in\overline{\Omega^{*}}\text{
near }y_{0}$

for some constant $\beta>1$, which in turn implies $u\in
C^{1,\alpha^{\prime}}$ as in part $ii)$ of Theorem 2.1.

In this paper, the notation $a\lesssim b$ (resp. $a\gtrsim b$) means that
there exists a constant $C>0$ independent of $h$ and the potential functions
$u$ and $v$, such that $a\leq Cb$ (resp. $a\geq Cb$), and the notation
$a\approx b$ means that $C^{-1}a\leq b\leq Ca$, where $a,b$ are both positive
constants. Given a convex domain $D\subset\mathbb{R}^{n}$, we say that $D$ has
a good shape if the eccentricity of its minimum ellipsoid is uniformly
bounded.

##  3\. A localisation lemma for $v$

###### Proof of Theorem 1.1.

Suppose $\partial_{nt}\Omega$ is not empty, and let $x_{0}$ be a point in
$\partial_{nt}\Omega.$ Denote $y_{0}=Du(x_{0}).$ Without loss of generality we
may assume $x_{0}=0.$ Denote $\nu=\frac{Du(0)}{|Du(0)|}.$ Up to a rotation of
coordinates we may assume $\nu=e_{n}$ is the $n$-th coordinate direction. By
the definition of $\partial_{nt}\Omega$ we have that

(3.1) $\Omega\subset\\{x_{n}\leq 0\\}.$

Let $G_{\epsilon}:=B_{\epsilon}(y_{0})\cap\Omega^{*},$ where $\epsilon$ is a
small positive number to be determined later. For any $p\in G_{\epsilon},$ let
$q=Dv(p).$ By the interior ball property, we have

(3.2) $|p-q|\leq|p-0|,$

since otherwise $0\in B_{|p-q|}(p)\cap\Omega\subset U$ contradicts to the
assumption that $0\in\partial\Omega\cap\overline{\Omega\cap\partial U}.$ By
(3.1) and (3.2) we have that

$\displaystyle|q_{n}|$ $\displaystyle\leq$ $\displaystyle|p-0|-p_{n}$
$\displaystyle=$ $\displaystyle
p_{n}\left(1+\frac{\sum_{i=1}^{n-1}p_{i}^{2}}{p^{2}_{n}}\right)^{1/2}-p_{n}$
$\displaystyle=$
$\displaystyle\frac{\sum_{i=1}^{n-1}p_{i}^{2}}{2p_{n}}+o(\frac{\sum_{i=1}^{n-1}p_{i}^{2}}{2p_{n}}).$

Then, it follows from the definition of $G_{\epsilon}$ and the above estimate
that

(3.3) $|q_{n}|\leq C\epsilon^{2}.$

By (3.2) again, we have that

$\sum_{i=1}^{n}|p_{i}-q_{i}|^{2}\leq\sum_{i=1}^{n}p_{i}^{2}.$

This implies

$\sum_{i=1}^{n}q_{i}^{2}\leq\sum_{i=1}^{n}2p_{i}q_{i}\leq\sum_{i=1}^{n-1}2p_{i}q_{i},$

where we have used the fact that $p_{n}>0,q_{n}<0.$ Hence, by Cauchy-Schwarz
inequality we have

$\sum_{i=1}^{n-1}q_{i}^{2}\leq
2(\sum_{i=1}^{n-1}p_{i}^{2})^{\frac{1}{2}}(\sum_{i=1}^{n-1}q_{i}^{2})^{\frac{1}{2}},$

which implies $\sum_{i=1}^{n-1}q_{i}^{2}\leq 4\sum_{i=1}^{n-1}p_{i}^{2}\leq
4\epsilon^{2}.$ Hence

(3.4) $|q_{i}|\leq 2\epsilon\ \ \text{ for}\ i=1,\cdots,n-1.$

By (3.3) and (3.4) we have that

(3.5) $|Dv(G_{\epsilon})|\leq C\epsilon^{n+1}.$

On the other hand, since $\Omega^{*}$ is convex, by the definition of
$G_{\epsilon},$ we have that $|G_{\epsilon}|\gtrsim\epsilon^{n}.$ Hence
$|G_{\epsilon}|\gg|Dv(G_{\epsilon})|$ as $\epsilon$ is sufficiently small,
which contradicts to the fact that $Dv$ is the optimal transport map between
$U$ and $\Omega^{*}$ with densities bounded between $1/\lambda$ and $\lambda$
for some positive constant $\lambda.$ Therefore, $\partial_{nt}\Omega$ must be
an empty set. Finally, by a standard covering argument we have that $U$ is
globally Lipschitz. ∎

Combining Theorem 1.1 and [5] we have the following useful localisation lemma
for $v.$

######  Lemma 3.1.

There exists $h_{0}>0$ small such that for any $y\in\overline{\Omega^{*}}$ we
have $S_{h}^{c}[v](y)\cap\overline{\Omega}=\emptyset,$ provided $h\leq h_{0}.$

Note that Lemma 3.1 is a strengthened version of (2.10). Then by Theorem 1.1
and [4, 9] we have the following important properties of $v.$

######  Lemma 3.2 (Uniform density).

Let $\Omega,\Omega^{*},f,g$ be as in Theorem 1.1. Assume that $\Omega^{*}$ is
$C^{2}$ and uniformly convex. For any $y_{0}\in\partial\Omega^{*},$ we have

(3.6)
${\small{\text{$\frac{|S_{h}^{c}[v](y_{0})\cap\Omega^{*}|}{|S_{h}^{c}[v](y_{0})|}$}}}\geq\delta,$

provided $0<h\leq h_{0},$ where $\delta$ is a positive constant depending on
$n,\lambda,\Omega^{*},$ but independent of $h$.

######  Corollary 3.1.

Under the conditions in Lemma 3.2, we have

  * $(i)$

_Volume estimate:_

(3.7)
$|S_{h}[v](y_{0})|\approx|S_{h}^{c}[v](y_{0})\cap\Omega^{*}|\approx|S_{h}^{c}[v](y_{0})|\approx
h^{\frac{n}{2}}.$

Moreover, for any given affine transform $\mathcal{A}$, if one of
$\mathcal{A}(S_{h}^{c}[v](y_{0}))$ and $\mathcal{A}(S_{h}[v](y_{0}))$ has a
good shape, so is the other one.

  * $(ii)$

_Tangential $C^{1,1-\epsilon}$ regularity for $v$:_ Assume in addition that
$f\in C(\overline{\Omega}),\ g\in C(\overline{\Omega^{*}})$. Let $\mathcal{H}$
be the tangent hyperplane of $\partial\Omega^{*}$ at $y_{0}$. Then
$\forall\,\epsilon>0$, $\exists\,C_{\epsilon}$ such that

(3.8) $B_{C_{\epsilon}h^{\frac{1}{2}+\epsilon}}(y_{0})\cap\mathcal{H}\subset
S_{h}^{c}[v](y_{0})\ \ \ \text{for $h>0$ small.}$

###### Proof.

Assume $y_{0}=0$ and write $S_{h}^{c}[v](0),S_{h}[v](0)$ as
$S_{h}^{c}[v],S_{h}[v]$ for brevity. By the strict convexity estimate of $v$
in $\overline{\Omega^{*}}$ (see (2.11)) and the fact that $S^{c}_{h}[v]$ is
balanced around $0$, we have an equivalence relation between $S_{h}[v]$ and
$S^{c}_{h}[v]$:

(3.9) $S^{c}_{b^{-1}h}[v]\cap\Omega^{*}\subset S_{h}[v]\subset
S^{c}_{bh}[v]\cap\Omega^{*}\ \ \ \forall\,h>0\text{ small},$

where $b\geq 1$ is a constant independent of $h$. For a proof of (3.9), we
refer the reader to [7, Lemma 2.2].

From Lemma 3.2 and (3.9), the volume estimate (3.7) can be deduced similarly
as in [4, Corollary 3.1]. Note that by (2.10) we have that
$\mbox{det}\,D^{2}v=\tilde{f}(y)\chi_{S^{c}_{h}[v]\cap\Omega^{*}}$ in
$S_{h}^{c}[v],$ where $\tilde{f}(y)=\frac{g(y)}{f(Dv(y))}\in
C(S^{c}_{h}[v]\cap\overline{\Omega^{*}}).$ Then, the proof of tangential
$C^{1,1-\epsilon}$ estimate is the same as in [4, Lemma 4.1]. ∎

##  4\. Obliqueness at intersection points

In this section we prove the obliqueness estimate at the intersection points
of the free boundary $\overline{\mathcal{F}}$ and the fixed boundary
$\partial\Omega$. Given $y_{0}\in\partial\Omega^{*},$ let $x_{0}:=Dv(y_{0}).$
If $x_{0}\in\partial\Omega\setminus\overline{\mathcal{F}}$ (resp.
$x_{0}\in\mathcal{F}$) the obliqueness estimate has been established in [7]
(resp. [9]). The situation becomes more complicated when
$x_{0}\in\partial\Omega\cap\overline{\mathcal{F}}.$ Denote by
$\nu_{\Omega}(x_{0}),\nu_{\Omega^{*}}(y_{0})$ the inner unit normal of
$\partial\Omega,\partial\Omega^{*}$ at $x_{0},y_{0}$ respectively, and denote
by
$\nu_{\overline{\mathcal{F}}}(x_{0}):=\frac{Du(x_{0})-x_{0}}{|Du(x_{0})-x_{0}|}$
the unit normal of $\overline{\mathcal{F}}$ at $x_{0}$ in the direction of
transportation. We have the following key estimate of this work.

######  Proposition 4.1.

Suppose the hypotheses of Theorem 1.2. Let $y_{0}\in\partial\Omega^{*},$ and
$x_{0}:=Dv(x_{0})\in\partial\Omega\cap\overline{\mathcal{F}}.$ Then
$\nu_{\Omega^{*}}(y_{0})\cdot\nu_{\overline{\mathcal{F}}}(x_{0})>0$ and
$\nu_{\Omega^{*}}(y_{0})\cdot\nu_{\Omega}(x_{0})>0.$

We need to rule out two cases: (i)
$\nu_{\Omega^{*}}(y_{0})\cdot\nu_{\overline{\mathcal{F}}}(x_{0})=0$; and (ii)
$\nu_{\Omega^{*}}(y_{0})\cdot\nu_{\Omega}(x_{0})=0.$

###  4.1. Case (i)

By a translation of coordinates we may assume $x_{0}=0.$ Denote by $e_{i}$,
$i=1,\cdots,n$ the standard coordinate directions of $\mathbb{R}^{n}.$ Up to a
rotation of coordinates we may assume $\nu_{\overline{\mathcal{F}}}(0)=e_{n}$
and $\nu_{\Omega^{*}}(y_{0})=e_{1}.$ Denote by $H_{\mathcal{F}}(0)$ the
tangent hyperplane of $\overline{\mathcal{F}}$ at $0.$ Denote by
$H_{\Omega}(0)$ the tangent hyperplane of $\Omega$ at $0.$ Let
$H^{\prime}:=H_{\mathcal{F}}(0)\cap H_{\Omega}(0).$ By the interior ball
property, we have that

(4.1) $B_{|y_{0}|}(y_{0})\cap\Omega\subset U.$

Note that $H_{\mathcal{F}}(0)$ is also tangent to $\partial
B_{|y_{0}|}(y_{0}).$ By Theorem 1.1, we have

(4.2) $\nu_{\Omega}(0)\cdot\nu_{\overline{\mathcal{F}}}(0)>-1.$

We will prove the following lemma.

######  Lemma 4.1.

There exists a $n-2$ dimensional $C^{2}$ submanifold of $\mathbb{R}^{n}$
(denoted by $\mathcal{M}$) passing through $0,$ such that its tangent space at
$0$ is contained in $H^{\prime}$ and $\mathcal{M}\cap
B_{r_{0}}(0)\subset\overline{U}$ for some small $r_{0}>0.$

###### Proof.

If $\nu_{\Omega}(0)\cdot\nu_{\overline{\mathcal{F}}}(0)=1,$ since
$\partial\Omega$ is $C^{2},$ by (4.2) we have that
$B_{r_{0}}(0)\cap\\{x_{n}>C|x^{\prime}|^{2}\\}\subset U$ for some small
positive constant $r_{0}$ and some large constant $C$ depending only the
$C^{2}$ norm of $\partial\Omega$ and the distance between $\Omega$ and
$\Omega^{*}.$ Hence we can take $\mathcal{M}=\\{x_{n}=C|x^{\prime}|^{2}\\}.$

If $-1<\nu_{\Omega}(0)\cdot\nu_{\overline{\mathcal{F}}}(0)<1,$ since both
$\partial\Omega$ and $\partial B_{|y_{0}|}(y_{0})$ are $C^{2},$ by implicit
function theorem we have that $\mathcal{M}:=\partial\Omega\cap\partial
B_{|y_{0}|}(y_{0})$ is an $n-2$ dimensional $C^{2}$ submanifold of
$\mathbb{R}^{n}.$ By (4.1) we have that $\mathcal{M}\cap
B_{r_{0}}(0)\subset\overline{U}$ for a small $r_{0}>0$.

Therefore, we can always find the desired $\mathcal{M}$ as in the statement of
the lemma. ∎

By subtracting a constant we can also assume that $v\geq 0$ and $v(y_{0})=0$.
Since $\overline{\Omega}\cap\overline{\Omega^{*}}=\emptyset,$ we have
$y_{0}=re_{n}$ for some $r>0$. Let $p=(p_{1},0,\cdots,0,p_{n})$ be a point on
$\partial\\{v<h\\}\cap\partial\Omega^{*}$ with $p_{n}<r.$ Denote $s=r-p_{n}$.
Since $\partial\Omega^{*}$ is $C^{2}$ smooth and uniformly convex, we have
$p_{1}=as^{2}+o(s^{2})$ for a positive constant $a.$ Similar to [9, Lemma
5.9], we have the following estimate for $s.$

######  Lemma 4.2.

For any $\epsilon>0$ small, there exist constants $C,C_{\epsilon}$ such that

(4.3) $Ch^{\frac{1}{3}}\leq s\leq C_{\epsilon}h^{\frac{1}{3}-\epsilon}$

when $h>0$ is small, where $C>0$ is a constant independent of $\varepsilon$.

###### Proof.

The proof is a small modification of that of [9, Lemma 5.9]. We explain the
necessary change here. For the second inequality, the proof is exactly same as
that of [9, Lemma 5.9]. It suffices to show the first inequality of (4.3).
Since $v\in C^{1}(\mathbb{R}^{n})$ and $Dv=\text{Id}$ in $\Omega\setminus U$.
Hence, as $0\in\mathcal{F}\subset\partial U$,

$Dv(0)=0=Dv(y_{0}).$

By the convexity of $v$, we infer that

$Dv(te_{n})=0\quad\forall\,t\in[0,r].$

Since $v(y_{0})=0$ and $v\geq 0$ on $\mathbb{R}^{n}$. Then $v(te_{n})=0$ for
all $t\in[0,r]$ as well. In particular, we have $v(z)=0$, where $z=p_{n}e_{n}$
is the projection of $p$ on the $x_{n}$ axis. Denote
$q=(q_{1},\cdots,q_{n})=Dv(p)\in\partial U$. By the convexity of $v$, we have

(4.4) $q_{1}=Dv(p)\cdot e_{1}\geq\frac{v(p)-v(z)}{|p-z|}=\frac{h}{p_{1}}\geq
C\frac{h}{s^{2}}.$

By the interior ball property (Lemma 2.1), we have
$B_{|p-q|}(p)\cap\Omega\subset U$. Hence

(4.5) $|p-q|^{2}\leq|p-0|^{2}.$

By monotonicity of $Dv,$ we have that $(p-y_{0})\cdot(q-0)\geq 0,$ which
implies

(4.6) $q_{n}\leq\frac{1}{s}p_{1}q_{1}.$

Note that this part is different from that in the proof of [9, Lemma 5.9].

Denote $q^{\prime}=(q_{1},q_{2},\cdots,q_{n-1})$. Recall that $p_{n}=r-s$. By
(4.5) and (4.6), we have

$|q^{\prime}|^{2}+p_{1}^{2}-2p_{1}q_{1}+\big{(}r-s-\frac{1}{s}p_{1}q_{1})\big{)}^{2}\leq
p_{1}^{2}+(r-s)^{2},$

from which one infers that $\frac{s}{r}|q^{\prime}|^{2}\leq 2p_{1}q_{1}$.
Noting that $q_{1}\leq|q^{\prime}|$, we thus obtain

$\frac{s}{r}q_{1}\leq 2p_{1}.$

Recall that $p_{1}\leq Cs^{2}+o(s^{2})$. By (4.4), we then deduce

$\frac{h}{sr}\leq Cp_{1}\leq Cs^{2},$

from which it follows that $s\geq Ch^{\frac{1}{3}}$. So the first inequality
of (4.3) is proved. ∎

Up to a translation of coordinates we may also assume $y_{0}=0.$ By
subtracting a constant we may assume $v(0)=0,v\geq 0.$ Since
$\nu_{\Omega^{*}}(0)\cdot\nu_{\overline{\mathcal{F}}}(0)=0,$ we can apply the
argument in [9, Section 5.2] to show that $S_{h}^{c}[v](0)$ can be normalised
by an affine transformation $T_{h},$ namely, $T_{h}(S_{h}^{c}[v])\sim
B_{1}(0).$ Moreover, $T_{h}=T_{2}\circ T_{1},$ where $T_{2}$ is an affine
transform satisfying

(4.7) $\|T_{2}\|+\|T_{2}^{-1}\|\leq C_{\epsilon}h^{-\epsilon}\quad\mbox{ for
any }\epsilon>0,$

and $T_{1}:y\mapsto\bar{y}$ is the transform given by

(4.8) $\left\\{{\begin{array}[]{ll}\bar{y}_{1}=h^{-\frac{2}{3}}y_{1},&\\\
\bar{y}_{i}=h^{-\frac{1}{2}}y_{i},&\ \ \ i=2,\cdots,n-1,\\\
\bar{y}_{n}=h^{-\frac{1}{3}}y_{n}.&\end{array}}\right.$

If $\nu_{\Omega}(0)\cdot\nu_{\overline{\mathcal{F}}}(0)=1,$ by the uniform
convexity and $C^{2}$ regularity of $\partial\Omega$, and the interior ball
property we have that $\partial U=\\{x_{n}=\rho(x^{\prime})\\}$ near 0 for
some function $\rho$ satisfying
$C^{-1}|x^{\prime}|^{2}\leq\rho(x^{\prime})\leq C|x^{\prime}|^{2},$ then we
can use the blow up argument in [9, Section 5] to make a contradiction.

In the following we only need to consider the situation
$-1<\nu_{\Omega}(0)\cdot\nu_{\overline{\mathcal{F}}}(0)<1,$ in which
$H^{\prime}:=H_{\mathcal{F}}(0)\cap H_{\Omega}(0)$ is an $n-2$-dimensional
subspace of $\mathbb{R}^{n}.$ We need to consider two subcases: 1) $e_{1}\in
H^{\prime};$ and 2) $e_{1}\notin H^{\prime}.$

####  4.1.1. Subcase 1: $e_{1}\in H^{\prime}$

In this subcase up a change of of $e_{2},\cdots,e_{n-1}$ coordinates we may
assume $H^{\prime}=\text{span}\\{e_{1},e_{3},\cdots,e_{n-1}\\}.$ For any unit
vector $e\in\text{span}\\{e_{2},\cdots,e_{n-1}\\},$ let
$p_{h}:=h^{\frac{1}{2}-3\epsilon}e+\rho^{*}(h^{\frac{1}{2}-3\epsilon}e)e_{1}\in\partial\Omega^{*},$
where $\Omega^{*}=\\{y_{1}>\rho^{*}(y_{2},\cdots,y_{n})\\}$ near $0$ for some
$C^{2},$ uniformly convex function $\rho^{*}$ with
$\rho^{*}(0)=0,D\rho^{*}(0)=0.$ A direct computation shows that

(4.9) $|T_{h}p_{h}-T_{h}(h^{\frac{1}{2}-3\epsilon}e)|\rightarrow 0,\
\text{as}\ h\rightarrow 0,$

and that

(4.10) $|T_{h}(h^{\frac{1}{2}-3\epsilon}e)|\rightarrow\infty,\ \text{as}\
h\rightarrow 0.$

By the Blaschke selection theorem and the standard technique of taking
diagonal sequences, we have that the convex sets $T_{h}(\Omega^{*})$ locally
uniformly converges to a limit convex set $\Omega^{*}_{0}$ in Hausdorff
distance. Up to a subsequence we may also assume that
$T_{h}(\text{span}\\{e_{2},\cdots,e_{n-1}\\})$ converges to $H^{*},$ which is
an $n-2$ dimensional subspace of $\mathbb{R}^{n}.$ By (4.9) and (4.10), we
have that $H^{*}\subset\overline{\Omega^{*}_{0}}.$ Hence by convexity we have
that the convex set $\Omega^{*}_{0}$ splits, namely,

(4.11) $\Omega^{*}_{0}=H^{*}\times\omega^{*}$

for some two dimensional convex set $\omega^{*}.$ Let

(4.12) $v_{h}(y):=\frac{1}{h}v(T_{h}^{-1}y).$

Then, $v_{h}$ is locally uniformly bounded in $\mathbb{R}^{n}$ as $h\to 0$.
Hence by passing to a subsequence, $v_{h}\rightarrow v_{0}$, locally
uniformly, and $v_{0}$ satisfies

(4.13) $\mbox{det}\,D^{2}v_{0}=c_{0}\chi_{{}_{\Omega^{*}_{0}}}\quad\text{in
}\mathbb{R}^{n}$

for a constant $c_{0}>0$. Since $v_{0}$ is a convex function defined on entire
$\mathbb{R}^{n},$ we have that $U_{0},$ the interior of $\partial
v_{0}(\mathbb{R}^{n})$ is a convex set. Since $\Omega^{*}_{0}$ is convex, we
have that $c_{0}\chi_{{}_{\Omega^{*}_{0}}}$ is doubling for any convex set
centred at a point in $\overline{\Omega^{*}_{0}}.$ Hence, by Caffarelli’s
boundary regularity theory [3], we have that $v_{0}$ is $C^{1}$ and strictly
convex on $\overline{\Omega^{*}_{0}}.$

Denote $T_{h}^{*}:=\frac{1}{h}(T^{t}_{h})^{-1}$. A straightforward computation
shows that the unit inner normal of $\Omega_{h}:=T_{h}^{*}(\Omega)$ at $0$ is
given by

(4.14)
$\nu_{{}_{\Omega_{h}}}(0):=\frac{T_{h}\nu_{\Omega}(0)}{|T_{h}\nu_{\Omega}(0)|}.$

Since $-1<\nu_{\Omega}(0)\cdot\nu_{\overline{\mathcal{F}}}(0)<1,$ we have that
$\nu_{\Omega}(0)=(0,c_{2},0,\cdots,0,c_{n})$ for some constants
$c_{2},c_{n}\neq 0$ satisfying $c_{2}^{2}+c_{n}^{2}=1.$ By the formula of
$T_{1},$ we have that
$T_{1}\nu_{\Omega}(0)=c_{2}h^{-\frac{1}{2}}e_{2}+c_{n}h^{-\frac{1}{3}}e_{n}.$
By (4.7) we have that

(4.15)
$\frac{T_{h}\nu_{\Omega}(0)}{|T_{h}\nu_{\Omega}(0)|}=\frac{T_{2}(T_{1}\nu_{\Omega}(0))}{|T_{2}(T_{1}\nu_{\Omega}(0))|}=\frac{T_{2}e_{2}}{|T_{2}e_{2}|}+o(1).$

Note that $\nu_{{}_{\Omega_{h}}}(0)$ is the unit inner normal of $\Omega_{h}$
at $0.$ Up to a subsequence we may assume that
$\frac{T_{2}e_{2}}{|T_{2}e_{2}|}$ converges to some unit vector $e_{0}\in
H^{*}$ as $h\rightarrow 0.$ By (4.15) we have $\nu_{{}_{\Omega_{h}}}(0)$
converges to $e_{0}$ as $h\rightarrow 0.$ Since
$Dv_{h}(\mathbb{R}^{n})\subset\overline{\Omega_{h}},$ we have that
$Dv_{h}(x)\cdot\nu_{{}_{\Omega_{h}}}(0)\geq 0$ for any $x\in\mathbb{R}^{n}.$
Passing to the limit we have that $Dv_{0}(x)\cdot e_{0}\geq 0$ for any
$x\in\mathbb{R}^{n}.$ Hence $v_{0}$ is monotone increasing along $e_{0}$
direction. Since $v_{0}(0)=0,v_{0}\geq 0,$ we have that $v_{0}(-te_{0})=0$ for
any $t>0,$ contradicts to the strict convexity of $v_{0}$ on
$\overline{\Omega^{*}_{0}}.$

####  4.1.2. Subcase 2: $e_{1}\notin H^{\prime}$

For any unit vector $e\in H^{\prime},$ let $p_{t}=te$ be a point on
$H^{\prime}$ with $t\leq h^{\frac{1}{2}-3\epsilon}.$ By Lemma 4.1 we have that
for any $t>0$ small, there exists a point $q_{t}\in\mathcal{M}\cap
B_{r_{0}}(0)\subset\overline{U},$ such that $|q_{t}-p_{t}|\leq Ct^{2}\leq
Ch^{1-6\epsilon}.$ Write

(4.16) $e=c_{1}e_{1}+c_{2}e_{2}+\cdots+c_{n-1}e_{n-1}.$

Since $e_{1}\notin H^{\prime},$ we have that at least one of
$c_{2},\cdots,c_{n-1}$ is not zero. Without loss of generality we may assume
$c_{2}\neq 0.$ Hence $|(T_{1}^{t})^{-1}p_{t}|\geq Ch^{-3\epsilon}.$ By (4.7)
we have that

(4.17) $|T^{*}_{h}p_{t}|\geq Ch^{-2\epsilon}\rightarrow\infty\ \text{as}\
h\rightarrow 0,$

where $T^{*}_{h}=\frac{1}{h}(T_{h}^{t})^{-1}$. By (4.8), (4.7) and (4.16) we
have that

(4.18) $|T^{*}_{h}q_{t}-T^{*}_{h}p_{t}|\leq
Ch^{\frac{1}{3}-6\epsilon}\rightarrow 0\ \text{as}\ h\rightarrow 0,$

provided $\epsilon$ is chosen small. Up to a subsequence we may assume
$T^{*}_{h}H^{\prime}$ converges $H,$ an $n-2$ dimensional subspace of
$\mathbb{R}^{n},$ locally uniformly as $h\rightarrow 0.$ By (4.17) and (4.18),
we have that $H\subset\overline{U_{0}}=\overline{Dv_{0}(\mathbb{R}^{n})}.$ By
convexity we also have that the convex set $U_{0}$ splits, namely,
$U_{0}=H\times\omega$ for some two dimensional convex set $\omega.$ Finally we
can apply the argument in [9] to arrive a contradiction.

###  4.2. Case (ii)

By a translation of coordinates we may assume $x_{0}=y_{0}=0.$ Denote by
$e_{i}$, $i=1,\cdots,n$, the standard coordinate directions of
$\mathbb{R}^{n}.$ Up to a rotation of coordinates we may assume
$\nu_{\Omega}(0)=e_{n}$ and $\nu_{\Omega^{*}}(0)=e_{1}$. Denote by
$H_{\mathcal{F}}(0)$ the tangent hyperplane of $\overline{\mathcal{F}}$ at
$0$. Denote by $H_{\Omega}(0):=\\{x_{n}=0\\}$ the tangent hyperplane of
$\Omega$ at $0$. Let $H^{\prime}:=H_{\mathcal{F}}(0)\cap H_{\Omega}(0),$ we
have that $H^{\prime}$ is an $(n-2)$-dimensional subspace of $\mathbb{R}^{n}$.
(Otherwise, if $H^{\prime}$ is $(n-1)$ dimensional, it goes back to Case (i).)
Observe that if $e_{1}\in H^{\prime},$ then
$e_{1}\cdot\nu_{\overline{\mathcal{F}}}(0)=0,$ which again can be reduced to
Case (i). Hence, in the following we always assume $e_{1}\notin H^{\prime}.$
Similar to the proof of [9, Lemma 5.20], up to an affine transformation we may
assume $H^{\prime}=\text{span}\\{e_{2},\cdots,e_{n-1}\\}.$ Now, we can carry
out a blow up argument developed in [6] as follows.

Let

$d_{e}:=\sup\\{|y\cdot e|:y\in S_{h}[v]\cap\Omega^{*}\\}$

for any unit vector $e.$ Denote $d_{e_{i}}$ by $d_{i}$ for short. Let
$p_{e}\in\overline{S_{h}[v]\cap\Omega_{1}^{*}}$ be the point such that
$p_{e}\cdot e=d_{e}.$ We have the following estimate.

######  Lemma 4.3.

$|d_{1}|\leq C_{\epsilon}h^{\frac{2}{3}-\epsilon},$ $d_{e}^{2}\leq Cd_{1}$ for
any unit vector $e\in\text{span}\\{e_{2},\cdots,e_{n}\\},$ where $\epsilon$
can be as small as we want.

###### Proof.

Suppose $d_{1}=p_{e_{1}}\cdot e_{1}\geq h^{\frac{2}{3}-4n\epsilon}.$ Let $q$
be the intersection of the ray $\\{p_{e_{1}}-te_{n}:t\geq 0\\}$ and
$\partial\Omega^{*},$ we have $q_{1}=d_{1}\geq h^{\frac{2}{3}-4n\epsilon}.$
Since $Dv(\mathbb{R}^{n})\subset\overline{\Omega}\subset\\{x_{n}\geq 0\\},$ we
have that $v$ is increasing in $e_{n}$ direction. Hence $v(q)\leq
v(p_{e_{1}})=h,$ which implies that $q\in\overline{S_{h}[v]}.$

Denote $e_{q}:=\frac{q-y_{0}}{|q-y_{0}|}.$ Denote by $D$ the planar region in
$\text{span}\\{e_{q},e_{1}\\},$ enclosed by
$\partial\Omega_{1}^{*}\cap\text{span}\\{e_{q},e_{1}\\}$ and the segment
$y_{0}q.$ By the $C^{2}$ regularity and uniform convexity of
$\partial\Omega_{1}^{*}$ we have that $\mathcal{H}^{2}(D)\geq
Cd_{1}^{\frac{3}{2}}.$ Let $\tilde{e}_{2},\cdots,\tilde{e}_{n-1}$ be an
orthonormal basis of the orthogonal complement of
$\text{span}\\{e_{1},e_{p}\\}$ in $\mathbb{R}^{n}.$ By the tangential
$C^{1,1-\epsilon}$ estimate of $v$ we have that
$y_{0}+h^{\frac{1}{2}+\epsilon}\tilde{e}_{i}\in S^{c}_{bh}[v]$,
$i=2,\cdots,n-1.$ Let $G$ be the convex envelope of $D$ and the points
$y_{0}+h^{\frac{1}{2}+\epsilon}\tilde{e}_{i}$, $i=2,\cdots,n-1.$ By convexity
we have $G\subset S^{c}_{bh}[v].$ Hence

$h^{(\frac{1}{2}+\epsilon)(n-2)}\mathcal{H}^{2}(D)\leq|G|\leq|S^{c}_{bh}[v]|\approx
h^{\frac{n}{2}},$

which implies that $|d_{1}|\leq C_{\epsilon}h^{\frac{2}{3}-\epsilon}.$

Fix any unit vector $e\in\text{span}\\{e_{2},\cdots,e_{n}\\},$ by the uniform
convexity of $\partial\Omega_{1}^{*}$ we have that $d_{1}\geq p_{e}\cdot
e_{1}\geq C(p_{e}\cdot e)^{2},$ which implies $d_{e}^{2}\leq Cd_{1}.$ ∎

First blow up. Suppose $S_{h}^{c}[v]\sim E$ for some ellipsoid centred at $0.$
Then, $E\cap\\{x_{1}=0\\}$ is an $(n-1)$-dimensional ellipsoid with principal
directions $\bar{e}_{2},\cdots,\bar{e}_{n}.$ Note that
$\text{span}\\{\bar{e}_{2},\cdots,\bar{e}_{n}\\}=\text{span}\\{e_{2},\cdots,e_{n}\\}.$
Now, we can write

$E=\left\\{x=x_{1}e_{1}+\sum_{i=2}^{n}\bar{x}_{i}\bar{e}_{i}:\frac{x_{1}^{2}}{a_{1}^{2}}+\sum_{i=2}^{n}\frac{(\bar{x}_{i}-k_{i}x_{1})^{2}}{a_{i}^{2}}\leq
1\right\\}.$

By Lemma 4.3 and the tangential $C^{1,1-\epsilon}$ estimate of $v$ at $y_{0},$
we have that

(4.19) $0<a_{1}<C_{\epsilon}h^{\frac{2}{3}-\epsilon},\ \text{and}\
C_{\epsilon}h^{\frac{1}{2}+\epsilon}<a_{i}<C_{\epsilon}h^{\frac{1}{3}-\epsilon}\
\text{for}\ i=2,\cdots,n.$

Let $\tilde{d}_{e}:=\\{|x\cdot e|:x\in S_{h}^{c}[v]\\}.$ By uniform density
and Lemma 4.3 we have

(4.20) $\tilde{d}_{e}^{2}\leq Ca_{1}\ \text{for any}\
e\in\text{span}\\{e_{2},\cdots,e_{n}\\}.$

It follows that

(4.21) $|k_{i}|\leq C\frac{a_{1}^{\frac{1}{2}}}{a_{1}}=Ca_{1}^{-\frac{1}{2}}.$

Let $T_{1},T_{2}$ be the affine transformations as following.

$T_{1}:x=x_{1}e_{1}+\sum_{i=2}^{n}\bar{x}_{i}\bar{e}_{i}\mapsto
z=x_{1}e_{1}+\sum_{i=2}^{n}(\bar{x}_{i}-k_{i}x_{1})\bar{e}_{i};$
$T_{2}:z=z_{1}e_{1}+\sum_{i=2}^{n}\bar{z}_{i}\bar{e}_{i}\mapsto
y=\frac{z_{1}}{a_{1}}e_{1}+\sum_{i=2}^{n}\frac{\bar{z}_{i}}{a_{i}}\bar{e}_{i}.$

Then $T_{h}(E)=B_{1},$ where $T_{h}=T_{2}T_{1}.$ Hence $T_{h}S_{h}^{c}[v]\sim
B_{1}.$ Let $v_{h}(x):=\frac{1}{h}v(T_{h}^{-1}x).$ Then
$Dv_{h}(x)=T_{h}^{*}Dv(T_{h}^{-1}x),$ where
$T_{h}^{*}=\frac{(T_{h}^{t})^{-1}}{h}=\frac{1}{h}(T_{2}^{t})^{-1}(T_{1}^{t})^{-1}.$
A direct computation shows that

$(T^{t}_{1})^{-1}:x=x_{1}e_{1}+\sum_{i=2}^{n}\bar{x}_{i}\bar{e}_{i}\mapsto
z=(x_{1}+\sum_{i=2}^{n}k_{i}\bar{x}_{i})e_{1}+\sum_{i=2}^{n}\bar{x}_{i}\bar{e}_{i};$
$(T^{t}_{2})^{-1}:z=z_{1}e_{1}+\sum_{i=2}^{n}\bar{z}_{i}\bar{e}_{i}\mapsto
y=a_{1}z_{1}e_{1}+\sum_{i=2}^{n}a_{i}\bar{z}_{i}\bar{e}_{i}.$

Let $\mathcal{M}$ be as in Lemma 4.1. For any given unit vector
$e\in\text{span}\\{e_{2},\cdots,e_{n-1}\\},$ Let $t_{h}$ be the positive
number such that $|\frac{1}{h}(T_{2}^{t})^{-1}(t_{h}e)|=1,$ we have that

$t_{h}\leq\frac{h}{\min_{2\leq i\leq n}a_{i}}\leq
C_{\epsilon}h^{\frac{1}{2}-\epsilon}.$

For any $t<h^{-\epsilon}t_{h},$ denote
$p_{t}:=te=(0,x^{t}_{2},\cdots,x^{t}_{n-1},0).$ It is straightforward to check
that

(4.22) $|T_{h}^{*}(h^{-\epsilon}t_{h}e)|\rightarrow\infty,\ \text{as}\
h\rightarrow 0.$

By Lemma 4.1, we can find
$q_{t}=p_{t}+x^{t}_{1}e_{1}+x^{t}_{n}e_{n}\in\mathcal{M}\cap
B_{r_{0}}(0)\subset\overline{U},$ with

(4.23) $|x^{t}_{1}|,\ |x^{t}_{n}|\leq Ct^{2}\leq Ch^{1-4\epsilon}.$

Let $z_{i}=e_{n}\cdot\bar{e}_{i},$ we have that
$e_{n}=\sum_{i=2}^{n}z_{i}\bar{e}_{i}.$ Then

$T_{h}^{*}(x^{t}_{1}e_{1}+x^{t}_{n}e_{n})=\frac{1}{h}x^{t}_{n}\left(a_{1}(\sum_{i=2}^{n}k_{i}z_{i})e_{1}+\sum_{i=2}^{n}a_{i}z_{i}\bar{e}_{i}\right)+\frac{1}{h}a_{1}x^{t}_{1}e_{1}.$

Since $|z_{i}|\leq 1$ for $i=2,\cdots,n-1,$ by (4.19), (4.21) and (4.23), a
straightforward computation shows that

$|T_{h}^{*}(x^{t}_{1}e_{1}+x^{t}_{n}e_{n})|\leq
Ch^{-4\epsilon}(h^{\frac{1}{3}-\frac{\epsilon}{2}}+h^{\frac{1}{3}-\epsilon})+h^{-4\epsilon}h^{\frac{2}{3}-\epsilon}\leq
Ch^{\frac{1}{3}-5\epsilon}.$

Hence,

(4.24) $|T^{*}_{h}(q_{t}-p_{t})|\rightarrow 0\ \text{ as }\ h\rightarrow 0,$

provided $\epsilon$ is sufficiently small.

Let $v_{h}$ be as in (4.12). Up to a subsequence we may assume that
$T_{h}(\Omega^{*}_{h})$ converges to a convex set $\Omega^{*}_{0}$ locally
uniformly as $h\rightarrow 0,$ and that $v_{h}$ converges to a convex function
$v_{0}$ locally uniformly as $h\rightarrow 0,$ and $v_{0}$ satisfies (4.13).
Let $U_{0}$ be the interior of the convex set $\overline{\partial
v_{0}(\mathbb{R}^{n})}.$ Up to a subsequence we may assume
$T_{h}^{*}(H^{\prime})$ converges to $H$ for some $n-2$ dimensional subspace
of $\mathbb{R}^{n}$ as $h\rightarrow 0.$ Note that
$e_{h}:=\frac{T_{h}e_{n}}{|T_{h}e_{n}|}$ is the unit inner normal of
$T_{h}^{*}(\Omega).$ Up to a subsequence we may assume that $e_{h}$ converges
to a unit vector $e_{0}.$ By the definition of $T_{h}$ we have that
$e_{h}\cdot e_{1}=0.$ Passing to limit we have that $e_{0}\cdot e_{1}=0.$
Since $T_{h}^{*}(H^{\prime})$ is a subspace of the tangent space of $\partial
T_{h}^{*}(\Omega)$ at $0,$ we have that $e_{h}$ is orthogonal to
$T_{h}^{*}(H^{\prime}),$ hence passing to limit we have that $e_{0}$ is
orthogonal to $H.$ Now, by a rotation of $e_{2},\cdots,e_{n}$ coordinates we
may assume $e_{0}=e_{n}$, $H=\text{span}\\{e_{2},\cdots,e_{n-1}\\}.$ By (4.22)
and (4.24) we have that $U_{0}$ splits, namely, $U_{0}=H\times\omega$ for some
two dimensional convex set $\omega$. Moreover $U_{0}\subset\\{x_{n}\geq 0\\}.$
Similar to the proof of [6, Lemma 3.5] we also have that $\Omega^{*}_{0}$ is a
smooth convex domain satisfying $\Omega^{*}_{0}=\\{x_{1}\geq
P(x_{2},\cdots,x_{n}\\}$ for some non-negative homogeneous quadratic
polynomial satisfying $P(0)=0,DP(0)=0.$

Finally, we can follow the argument in [6] to perform a second blow up and
then use the argument in [7] to deduce a contradiction.

##  5\. $C^{1,1-\epsilon}$ estimate

In this section we assume $\Omega,\Omega^{*},f,g$ satisfy the same conditions
as those in Proposition 4.1. We will establish the $C^{1,1-\epsilon}$ estimate
of $u.$ For any $y_{0}\in\partial\Omega^{*},$ denote
$x_{0}=Dv(y_{0})\in\partial U.$ In the following we will establish a pointwise
$C^{1,1-\epsilon}$ estimate of $v$ at $x_{0}.$ After a translation of
coordinates and subtracting an affine function to $v$, we may assume that
$x_{0}=y_{0}=0,$ and that $v(0)=0$, $v\geq 0.$

######  Theorem 5.1.

There exists a small constant $r>0$ independent of the location of $y_{0}=0$
on $\partial\Omega^{*},$ such that

$0\leq v(y)\leq C_{\epsilon}|y|^{2-\epsilon}$

for any $y\in B_{r}(0).$

We will show such estimate in the worst scenario when
$x_{0}\in\overline{\mathcal{F}}\cap\partial\Omega.$ Note that near such point,
$\partial U$ is only known to be Lipschitz, and it is also not known whether
$\partial U$ is locally convex. By Proposition 4.1, we can find an affine
transform $A$ such that $(A^{t})^{-1}\nu_{\Omega^{*}}(0)$ (the inner normal
direction of $A\Omega^{*}$ at $0$) is parallel to $A\nu_{\Omega}(0)$ (the
inner normal direction of $(A^{t})^{-1}\Omega$ at $0$). Hence up to an affine
transformation we may assume $\nu_{\Omega^{*}}(0)=\nu_{\Omega}(0)=e_{n}.$
Moreover, by Proposition 4.1 we also have that there exists constants $K,r>0$
such that

(5.1) $\mathcal{C}_{K,r}:=\\{x_{n}\geq K|x^{\prime}|\\}\cap B_{r}(0)\subset
U.$

Let $z_{h}=c_{h}e_{n}$ be the intersection of positive $x_{n}$ axis and
$\partial S_{h}^{c}[v].$ To prove Theorem 5.1 we only need to establish the
following lemma.

######  Lemma 5.1.

For any $\eta>0$ small, there exists a constant $C_{\eta},$ such that
$c_{h}>C_{\eta}h^{\frac{1}{2}+\eta}.$

###### Proof.

Suppose to the contrary that for some $\eta>0$ small one cannot find such
$C_{\eta},$ namely, for any given $c_{0}>0,$ there exists $h>0$ such that
$c_{h}\leq c_{0}h^{\frac{1}{2}+\eta}.$ Let $h_{0}:=\max\\{h:c_{h}\leq
c_{0}h^{\frac{1}{2}+\eta}.$ By taking $c_{0}\rightarrow 0,$ it is
straightforward to check that $h_{0}\rightarrow 0.$ Hence
$c_{h_{0}}=c_{0}h_{0}^{\frac{1}{2}+\eta}.$

Let $M$ be a large constant to be determined later. There exists a constant
$C$ depending only on $n,$ such that $S_{Mh_{0}}^{c}[v]\subset
CMS^{c}_{h_{0}}[v].$ Suppose $S_{Mh_{0}}^{c}[v]\sim E$ for some ellipsoid
centred at $0.$ Then, $E\cap\\{x_{n}=0\\}$ is an $(n-1)$-dimensional
ellipsoid. By a rotation of $x_{1},\cdots,x_{n-1}$ coordinates we may assume
that the principal directions of $E\cap\\{x_{n}=0\\}$ are
$e_{1},\cdots,e_{n-1}.$ We can write

$E=\left\\{\sum_{i=1}^{n}x_{i}e_{i}:\sum_{i=1}^{n-1}\frac{(x_{i}-k_{i}x_{1})^{2}}{a_{i}^{2}}+\frac{x_{n}^{2}}{a_{n}^{2}}\leq
1\right\\}.$

By tangential $C^{1,1-\epsilon}$ of $v$ we have that

(5.2) $a_{i}\geq C_{\epsilon}h_{0}^{\frac{1}{2}+\epsilon},\ \text{for}\
i=1,\cdots,n-1.$

Let $p_{h_{0}}=t_{h_{0}}e_{n}$ be the intersection of the positive $x_{n}$
axis and $\partial E.$ By the contradiction assumption we also have

(5.3) $t_{h_{0}}\geq C_{n}c_{0}(Mh_{0})^{\frac{1}{2}+\eta}.$

Let $E^{*}$ be the dual ellipsoid of $E,$ namely,

(5.4) $E^{*}:=\\{y:y\cdot x\leq h\ \text{for any}\ x\in E\\}.$

Then, $E^{*}\cap\\{x_{n}=0\\}$ is an $(n-1)$-dimensional ellipsoid with
principal directions $\bar{e}_{1},\cdots,\bar{e}_{n-1}.$ Note that
$\text{span}\\{\bar{e}_{1},\cdots,\bar{e}_{n-1}\\}=\text{span}\\{e_{1},\cdots,e_{n-1}\\}.$
We can now write

$E^{*}=\left\\{y=y_{n}e_{n}+\sum_{i=1}^{n-1}\bar{y}_{i}\bar{e}_{i}:\sum_{i=1}^{n-1}\frac{(\bar{y}_{i}-k^{\prime}_{i}y_{n})^{2}}{b_{i}^{2}}+\frac{y_{n}^{2}}{b_{n}^{2}}\leq
1\right\\}.$

By (5.3) and (5.4) we have that

(5.5) $b_{n}\geq\frac{C}{c_{0}}h_{0}^{\frac{1}{2}-\eta}.$

By (5.2) we have that for any $e\in\text{span}\\{e_{1},\cdots,e_{n-1}\\},$

(5.6) $\sup\\{y\cdot e:y\in
E^{*}\\}\leq\frac{1}{C_{\epsilon}}h_{0}^{\frac{1}{2}-\epsilon}\quad\forall\,\epsilon>0.$

Hence the following estimates holds:

(5.7)
$|k_{i}^{\prime}|\leq\frac{\frac{1}{C_{\epsilon}}h_{0}^{\frac{1}{2}-\epsilon}}{b_{n}}\leq\frac{\frac{1}{C_{\epsilon}}h_{0}^{\frac{1}{2}-\epsilon}}{\frac{C}{c_{0}}h_{0}^{\frac{1}{2}-\eta}}\leq
Ch_{0}^{\eta-\epsilon}\rightarrow 0\ \text{as}\ h_{0}\rightarrow 0,$

provided $\epsilon<\eta.$ By (5.6) we also have

$|b_{i}|\leq\frac{1}{C_{\epsilon}}h_{0}^{\frac{1}{2}-\epsilon}.$

Hence

(5.8) $\frac{b_{n}}{b_{i}}\geq
C_{\epsilon}h_{0}^{\epsilon-\eta}\rightarrow\infty\ \text{as}\
h_{0}\rightarrow 0\ \text{for}\ i=1,\cdots,n-1.$

Let $A_{h_{0}}$ be an affine transformation such that $A_{h_{0}}E=B_{1}(0),$
then $A_{h_{0}}^{*}(E^{*})=B_{1}(0),$ where
$A_{h_{0}}^{*}:=\frac{1}{h_{0}}(A_{h_{0}}^{t})^{-1}.$ Let $T_{h_{0}}$ be the
affine transform

$T_{h_{0}}:x=x_{1}e_{1}+\sum_{i=2}^{n}\bar{x}_{i}\bar{e}_{i}\mapsto
z=\frac{x_{n}}{b_{n}}e_{n}+\sum_{i=1}^{n-1}\frac{\bar{x}_{i}-k^{\prime}_{i}x_{n}}{b_{i}}\bar{e}_{i}.$

Then, $T_{h_{0}}E^{*}=B_{1}(0).$ Hence $A^{*}_{h_{0}}T_{h_{0}}^{-1}$ is an
affine transform satisfying $A^{*}_{h_{0}}T_{h_{0}}^{-1}(B_{1}(0))=B_{1}(0),$
which implies that $A^{*}_{h_{0}}T_{h_{0}}^{-1}$ is a linear isometry from
$\mathbb{R}^{n}$ to $\mathbb{R}^{n}.$ By (5.7) and (5.8), a direct computation
shows that $T_{h_{0}}\mathcal{C}_{K,r}$ converges to the upper half space
$\\{x_{n}\geq 0\\}$ locally uniformly in Hausdorff distance as $h\rightarrow
0.$ Since $A_{h_{0}}T_{h_{0}}^{-1}$ is a linear isometry, we have that

(5.9)
$A^{*}_{h_{0}}\mathcal{C}_{K,r}=A^{*}_{h_{0}}T_{h_{0}}^{-1}T_{h_{0}}\mathcal{C}_{K,r}\rightarrow\\{x\in\mathbb{R}^{n}:x\cdot
e_{0}\geq 0\\}$

locally uniformly in Hausdorff distance for some unit vector $e_{0},$ as
$h_{0}\rightarrow 0.$

Similar to the proof of [6, Lemma 3.5], up to a subsequence we may assume
$A_{h_{0}}(\Omega^{*})$ locally uniformly converges to a smooth convex set
$\Omega_{0}^{*}$ as $h_{0}\rightarrow 0.$ Let

(5.10) $v_{h_{0}}(y):=\frac{1}{Mh_{0}}v(A_{h_{0}}^{-1}y).$

Then, $v_{h}$ is locally uniformly bounded in $\mathbb{R}^{n}$ as $h\to 0$.
Hence by passing to a subsequence, $v_{h_{0}}\rightarrow v_{0}$, locally
uniformly, and $v_{0}$ satisfies

(5.11) $\mbox{det}\,D^{2}v_{0}=c_{0}\chi_{{}_{\Omega^{*}_{0}}}\quad\text{in
}\mathbb{R}^{n}$

for a constant $c_{0}>0$. Since $v_{0}$ is a convex function defined on entire
$\mathbb{R}^{n},$ we have that $U_{0},$ the interior of $\partial
v_{0}(\mathbb{R}^{n})$ is a convex set. By (5.9) we have that
$U_{0}=\\{x\in\mathbb{R}^{n}:x\cdot e_{0}>0\\},$ in particular $\partial
U_{0}$ is smooth. Since both $\Omega^{*}_{0}$ and $U_{0}$ are smooth convex,
$v_{0}$ is a convex solution to (5.11) satisfying $v_{0}(0)=0,v_{0}\geq 0,$
hence by the regularity theory of [7] we have that $0\leq v_{0}(x)\leq
C|x|^{2}$ for any $x\in\Omega_{0}^{*}.$ Hence
$B_{ch^{\frac{1}{2}}}(0)\cap\Omega^{*}_{0}\subset S_{h}[v_{0}].$ Since
$S_{\frac{1}{C}h}[v_{0}]\subset S_{h}^{c}[v_{0}]\cap\Omega^{*}_{0}$ and
$|S_{h}[v_{0}]|\approx h^{\frac{n}{2}},$ we have that $S_{h}^{c}[v_{0}]\sim
B_{h^{\frac{1}{2}}}(0).$ Since $v_{0}(y)\leq Ch$ for any $y\in
S_{h}^{c}[v_{0}],$ it follows that

(5.12) $v_{0}(y)\leq C|y|^{2}$

for any $y\in B_{1}(0).$

There exists a constant $C_{1}$ depending only on $n,$ such that
$-C_{1}c_{h_{0}}e_{n}\notin S^{c}_{h_{0}}[v].$ Let
$h=C_{1}c_{h_{0}}=C_{1}c_{0}h_{0}^{\frac{1}{2}+\eta}.$ Hence

(5.13) $\Delta_{h}v:=v(he_{n})+v(-he_{n})-2v(0)\geq 2h_{0}.$

Denote $e_{h_{0}}=A_{h_{0}}(t_{h_{0}}e_{n}).$ Then

$\displaystyle\Delta_{h}v$ $\displaystyle=$ $\displaystyle
Mh_{0}\left(v_{h_{0}}(t_{h_{0}}^{-1}he_{h_{0}})+v_{h_{0}}(-t_{h_{0}}^{-1}he_{h_{0}})-2v_{h_{0}}(0)\right)$
$\displaystyle\leq$ $\displaystyle
Mh_{0}\left(Ct_{h_{0}}^{-2}h^{2}+4\|v_{h_{0}}-v_{0}\|_{L^{\infty}(B_{1}(0))}\right)$
$\displaystyle\leq$ $\displaystyle
h_{0}\left(CM\frac{C_{1}^{2}c_{0}^{2}h_{0}^{1+2\eta}}{C_{n}^{2}c_{0}^{2}M^{1+2\eta}h_{0}^{1+2\eta}}+4\|v_{h_{0}}-v_{0}\|_{L^{\infty}(B_{1}(0))}\right)$
$\displaystyle=$ $\displaystyle
h_{0}\left(CC_{1}^{2}C_{n}^{-2}M^{-2\eta}+4\|v_{h_{0}}-v_{0}\|_{L^{\infty}(B_{1}(0))}\right).$

By first taking $M$ sufficiently large we can have
$CC_{1}^{2}C_{n}^{-2}M^{-2\eta}\leq\frac{1}{4}.$ Then by taking $h_{0}$
sufficiently small and we can have
$4\|v_{h_{0}}-v_{0}\|_{L^{\infty}(B_{1}(0))}\leq\frac{1}{4}.$ Now
$\Delta_{h}v\leq h_{0}(\frac{1}{4}+\frac{1}{4})=\frac{1}{2}h_{0},$
contradicting to (5.13).

Therefore, we can always find the desired $C_{\eta},$ such that
$c_{h}>C_{\eta}h^{\frac{1}{2}+\eta}.$ ∎

##  6\. $W^{2,p}$ estimate

By Theorem (5.1), we have that
$B_{C_{\epsilon}h^{\frac{1}{2}+\epsilon}}(0)\cap\Omega^{*}\subset S_{h}[v].$
Then, since $|S_{h}[v]|\approx h^{\frac{n}{2}},$ we have that

$B_{C_{\epsilon}h^{\frac{1}{2}+\epsilon}}(0)\cap\Omega^{*}\subset
S^{c}_{h}[v]\subset B_{C_{\epsilon}h^{\frac{1}{2}-\epsilon}}(0).$

This implies $v(y)\geq C_{\epsilon}|y|^{2+\epsilon}.$ Since $u=v^{*},$ we have
that $u(x)\leq C_{\epsilon}|x|^{2-\epsilon}$ for any $x\in\overline{U}.$
Hence, $u\in C^{1,1-\epsilon}(\overline{U}).$ The proof of Theorem 1.1 can
then be completed following the proof of [7, Theorem 1.2].

##  7\. Counter example for $C^{1,1}$ regularity

The $W^{2,p}$ regularity of $u$ in $\overline{U}$ is sharp in the sense that
we can construct examples showing that even assuming further that the
densities are smooth, the solution $u$ is not global $C^{1,1}$ in $U.$ The
examples can be constructed as follows. Let $p_{0}=(2,0)\in\mathbb{R}^{2}.$
Let $\Omega=B_{1}(0),\Omega^{*}=B_{\frac{1}{100}}(p_{0})$ and
$f=\chi_{\Omega},g=\chi_{\Omega^{*}}.$ Let $u$ be the solution to the obstacle
problem (1.1). First, we claim that

(7.1) $U\subset B_{1}(0)\cap\\{x_{1}>0\\}.$

Suppose not, then there exists $p\in U\cap\\{x_{1}\leq 0\\}.$ Denote
$q_{0}:=(1,0).$ Since
$\int_{B_{\frac{1}{10}}(q_{0})\cap\Omega}f>\int_{\Omega^{*}}g$ and

$\text{dist}(p,\Omega^{*})>\sup\\{|x-y|:x\in
B_{\frac{1}{10}}(q_{0})\cap\Omega,y\in\Omega^{*}\\},$

we can construct a cheaper transport plan by replacing the mass in $U$ near
$p$ by the mass in $\left(B_{\frac{1}{10}}(q_{0})\cap\Omega\right)\backslash
U,$ which is a contradiction. Let $x_{0}$ be a point in
$\overline{\mathcal{F}}\cap\partial\Omega.$ By (2.6) we have that
$\nu_{\overline{\mathcal{F}}}(x_{0})=\frac{Du(x_{0})-x_{0}}{|Du(x_{0})-x_{0}|}.$
Then by Theorem 1.1 and (7.1), it is straightforward to verify that
$-1<\nu_{\overline{\mathcal{F}}}(x_{0})\cdot\nu_{\Omega}(x_{0})<1.$

Now we adapt an argument of Savin and Yu [19, 20] to show that $u\notin
C^{1,1}(\overline{U}).$ Suppose to the contrary $u\in C^{1,1}(\overline{U}).$
By a translation of coordinates and subtracting an affine function, we may
assume $x_{0}=0,Du(x_{0})=0,u\geq 0.$ Up to an affine transform we may assume
$\nu_{\overline{\mathcal{F}}}(0)=\nu_{\Omega^{*}}(0)=e_{2}.$ Since
$\partial\Omega,\overline{\mathcal{F}}$ are all at least $C^{1}$ regular, then
$\lambda U$ converges to a cone
$\mathcal{C}:=\\{tz:t>0,z=(\cos\theta,\sin\theta),\theta\in(0,\theta_{0})\\}$
as $\lambda\rightarrow\infty,$ where $0<\theta_{0}<\pi.$ By Proposition 4.1 we
actually have $\frac{\pi}{2}<\theta_{0}<\pi.$ Since $\partial\Omega^{*}$ is
$C^{2}$ we have that $\lambda\Omega^{*}$ converges to $\\{x_{2}>0\\}$ as
$\lambda\rightarrow\infty.$ Now, let
$u_{h}(x):=\frac{1}{h}u(h^{\frac{1}{2}}x).$ Then up to a subsequence $u_{h}$
converges to a convex function $u_{0}$ locally uniformly as $h\rightarrow 0,$
with $u_{0}\in C^{1,1}(\mathcal{C})$ and $Du_{0}(\mathcal{C})=\\{x_{2}>0\\}.$
Let $v_{0}(y):=\sup_{x\in\mathcal{C}}x\cdot y-u_{0}(y),$ then $v_{0}\in
C^{1,1}(\\{x_{2}\geq 0\\}),$ $\mbox{det}D^{2}v_{0}=1$ in $\\{x_{2}>0\\}$ and
$Dv_{0}(\\{x_{2}>0\\})=\mathcal{C}.$

Fix any unit vector $e,$ let $p_{k}\in\\{x_{2}>0\\}$ be a sequence such that
$\partial_{{}_{ee}}v_{0}(p_{k})$ converges to
$\sup_{x\in\\{x_{2}>0\\}}\partial_{{}_{ee}}v_{0}(x).$ Let
$v_{0k}(x)=\frac{1}{|p_{k}|^{2}}v_{0}(|p_{k}|x).$ Note that quadratic
rescaling preserves second derivatives. By compactness, we may also assume
$\frac{p_{k}}{|p_{k}|}$ converges to a point $p_{\infty}\in\\{x_{2}\geq 0\\}$
as $k\rightarrow\infty.$ Note that $|p_{\infty}|=1.$ Up to a subsequence we
may assume $v_{0k}$ converges to a convex function $\tilde{v}_{0}$ locally
uniformly as $k\rightarrow\infty.$ Now $\tilde{v}_{0}$ satisfies
$\tilde{v}_{0}\in C^{1,1}(\\{x_{2}\geq 0\\}),$
$\mbox{det}D^{2}\tilde{v}_{0}=1$ in $\\{x_{2}>0\\}$ and
$D\tilde{v}_{0}(\\{x_{2}>0\\})=\mathcal{C}.$ Moreover, the maximum of
$\partial_{{}_{ee}}\tilde{v}_{0}$ is attained at $p_{\infty}\in\\{x_{2}\geq
0\\}.$ If $p_{\infty}\in\\{x_{2}>0\\},$ then by maximum principle we have
$\partial_{{}_{ee}}\tilde{v}_{0}$ is constant in $\\{x_{2}>0\\}.$ If
$p_{\infty}\in\\{x_{2}=0\\},$ since $|p_{\infty}|=1,$ we have
$p_{\infty}=(1,0)$ or $p_{\infty}=(-1,0).$ In either case, up to an affine
transform we may assume $D\tilde{v}_{0}\cdot e_{2}=0$ in
$B_{\frac{1}{2}}(p_{\infty})\cap\\{x_{2}=0\\}.$ Then we can extend
$\tilde{v}_{0}$ to the entire $B_{\frac{1}{2}}(p_{\infty})$ by reflection,
namely let $\tilde{v}_{0}(x_{1},x_{2})=\tilde{v}_{0}(x_{1},-x_{2})$ whenever
$x_{2}<0.$ Then $\tilde{v}_{0}$ is a strictly convex function in
$B_{\frac{1}{2}}(p_{\infty})$ satisfies $\mbox{det}D^{2}\tilde{v}_{0}=1$ in
$B_{\frac{1}{2}}(p_{\infty}).$ Now $\partial{{}_{ee}}\tilde{v}_{0}$ attains
its maximum in an interior point $p_{\infty},$ by maximum principle again we
have that $\partial_{{}_{ee}}\tilde{v}_{0}$ is constant in $\\{x_{2}>0\\}.$

Therefore, $\tilde{v}_{0}$ is a quadratic polynomial and hence
$D\tilde{v}_{0}$ maps half space to half space, which contradicts to
$D\tilde{v}_{0}(\\{x_{2}>0\\})=\mathcal{C}.$ Hence, the contradiction
assumption fails, namely $u\notin C^{1,1}(\overline{U}).$

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Polynomial Sequences Associated with the Moments of Hypergeometric Weights

Polynomial Sequences Associated with the Moments
of Hypergeometric Weights\star\star$\star$This paper is a contribution to the
Special Issue on Orthogonal Polynomials, Special Functions and Applications.
The full collection is available at
http://www.emis.de/journals/SIGMA/OPSFA2015.html

Diego DOMINICI

D. Dominici

Department of Mathematics, State University of New York at New Paltz,
1 Hawk Dr., New Paltz, NY 12561-2443, USA<EMAIL_ADDRESS>https://www2.newpaltz.edu/~dominicd/

Received November 23, 2015, in final form April 25, 2016; Published online
April 29, 2016

We present some families of polynomials related to the moments of weight
functions of hypergeometric type. We also consider different types of
generating functions, and give several examples.

moments; hypergeometric functions; generating functions; Stieltjes transform

44A60; 33C20; 05A15

Dedicated to my daughter Malena, luz de mi vida!

##  1 Introduction

Let $\\{\mu_{n}\\}$ be a sequence of complex numbers and
$\mathcal{L}\colon\mathbb{C}[x]\rightarrow\mathbb{C}$ be a linear functional
defined by

$\displaystyle\mathcal{L}\big{(}x^{n}\big{)}=\mu_{n},\qquad n=0,1,\ldots.$

Then, $\mathcal{L}$ is called the _moment functional_ determined by the formal
moment sequence $\\{\mu_{n}\\}$. The number $\mu_{n}$ is called the _moment_
of order $n$. The task of finding an explicit representation for the
functional $\mathcal{L}$ is called a _moment problem_ [1, 24, 32].

A sequence $\\{\Pi_{n}(x)\\}\subset\mathbb{C}[x]$, with $\deg(\Pi_{n})=n$ is
called an _orthogonal polynomial sequence_ with respect to $\mathcal{L}$
provided that [4]

$\displaystyle\mathcal{L}(\Pi_{n}\Pi_{m})=K_{n}\delta_{n,m},\qquad
n,m=0,1,\ldots,$

where $K_{n}\neq 0$ and $\delta_{n,m}$ is Kronecker’s delta.

The moments play a fundamental role in the theory of orthogonal polynomials
since, among other results, we have the determinantal representation

$\displaystyle\Pi_{n}(x)=C_{n}\left|\begin{matrix}\mu_{0}&\mu_{1}&\cdots&\mu_{n}\\\
\mu_{1}&\mu_{2}&\cdots&\mu_{n+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mu_{n-1}&\mu_{n}&\cdots&\mu_{2n-1}\\\ 1&x&\cdots&x^{n}\end{matrix}\right|,$

for some normalization constant $C_{n}\neq 0$, with

$\displaystyle\left|\begin{matrix}\mu_{0}&\mu_{1}&\cdots&\mu_{n}\\\
\mu_{1}&\mu_{2}&\cdots&\mu_{n+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
\mu_{n}&\mu_{n+1}&\cdots&\mu_{2n}\end{matrix}\right|\neq 0,\qquad
n=0,1,\ldots.$

Given their importance, it is very striking that they are not explicitly
listed in the standard books on orthogonal polynomials, or even in
encyclopedic texts such as [23]. In fact, the only place where we found a
comprehensive enumeration of the moments of classical orthogonal polynomials
was the recent article [28], based on the results obtained in the Ph.D. Thesis
of the first author [27]111We thank one of the referees for stressing this
fact..

In this paper, we focus our attention on linear functionals defined by

$\displaystyle\mathcal{L}(f)={\displaystyle\sum\limits_{x=0}^{\infty}}f(x)\rho\big{(}x;\vec{\alpha},\vec{\beta},c\big{)},$
(1)

where the weight function $\rho(x;\vec{\alpha},\vec{\beta},c)$ is of the form

$\displaystyle\rho\big{(}x;\vec{\alpha},\vec{\beta},c\big{)}=\frac{(\vec{\alpha})_{x}}{(\vec{\beta}+1)_{x}}\frac{c^{x}}{x!},$
(2)

with

$\displaystyle(\vec{\alpha})_{x}=(\alpha_{1})_{x}(\alpha_{2})_{x}\cdots(\alpha_{p})_{x},\qquad(\vec{\beta}+1)_{x}=(\beta_{1}+1)_{x}(\beta_{2}+1)_{x}\cdots(\beta_{q}+1)_{x},$

and $(a)_{x}$ denotes the Pochhammer symbol (also called shifted or rising
factorial) defined by [29, (5.2.4)]

$\displaystyle(a)_{0}=1,\qquad(a)_{x}=a(a+1)\cdots(a+x-1),\qquad
x\in\mathbb{N},$

or by

$\displaystyle(a)_{x}=\frac{\Gamma(a+x)}{\Gamma(a)},\qquad a+x\neq
0,-1,\ldots,$

and $\Gamma(z)$ is the Gamma function. Unless stated otherwise, we always
assume that

$\displaystyle\beta_{i}>-1,\qquad 1\leq i\leq q,$

and we will use the notation

$\displaystyle\vec{\alpha}=\alpha_{1},\alpha_{1},\ldots,\alpha_{p},\qquad\vec{\beta}+1=\beta_{1}+1,\beta_{2}+1,\ldots,\beta_{q}+1.$

Note that we have

$\displaystyle\frac{\rho(x;\vec{\alpha},\vec{\beta},c)}{\rho(x;\vec{\alpha},\vec{\beta},c)}=\frac{\eta(x)}{\phi(x+1)},$

with

$\displaystyle\phi(x)=x(x+\beta_{1})(x+\beta_{2})\cdots(x+\beta_{q}),\qquad\eta(x)=c(x+\alpha_{1})(x+\alpha_{2})\cdots(x+\alpha_{p}).$
(3)

Hence, the weight function $\rho(x;\vec{\alpha},\vec{\beta},c)$ satisfies the
_Pearson equation_ (see [30] or [31, (6.3)])

$\displaystyle\Delta(\phi\rho)=(\eta-\phi)\rho,$

where

$\displaystyle\Delta f(x)=f(x+1)-f(x)$

is the forward difference operator. If we define $s$ by

$\displaystyle s=\max\big{\\{}\deg(\phi)-2,\deg(\phi-\eta)-1\big{\\}},$

the sequence of polynomials orthogonal with respect to $\mathcal{L}$ is called
_semiclassical_ of class $s$ [18, 25].

Weight functions of the form (2) are also related to discrete probability
distributions (Poisson, Pascal, binomial, hypergeometric, etc.) [21]222We
thank one of the editors for suggesting this reference.. The Generalized
Hypergeometric probability distributions were studied by Adrienne W. Kemp in
her Ph.D. Thesis [22]333Unfortunately, we haven’t been able to obtain a copy
of it.. An excellent reference outlining the connections between the theory of
probability and orthogonal polynomials is [31].

In [5], it was pointed out that since

$\displaystyle\mu_{n}(c)=\sum\limits_{x=0}^{\infty}x^{n}\rho\big{(}x;\vec{\alpha},\vec{\beta},c\big{)},$
(4)

one has

$\displaystyle\mu_{n+1}(c)=\sum\limits_{x=0}^{\infty}x^{n}x\rho\big{(}x;\vec{\alpha},\vec{\beta},c\big{)}=\vartheta\mu_{n}(c),$
(5)

where the differential operator $\vartheta$ is defined by [29, (16.8.2)]

$\displaystyle\vartheta f(c)=c\frac{df}{dc}.$ (6)

Successive applications of (5) give

$\displaystyle\mu_{n}=\vartheta^{n}\mu_{0},$

and it follows that the first moment $\mu_{0}$ determines the whole sequence
$\\{\mu_{n}\\}$. If we use the operational formula [26]

$\displaystyle\vartheta^{k}=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}c^{k}\frac{d^{k}}{dc^{k}},$

we have

$\displaystyle\mu_{n}(c)=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}c^{k}\frac{d^{k}\mu_{0}}{dc^{k}},$
(7)

where $\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}$ denote the Stirling numbers of the
second kind defined by [29, (26.8)]

$\displaystyle\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}=\frac{1}{k!}{\displaystyle\sum\limits_{j=0}^{k}}\binom{k}{j}(-1)^{k-j}j^{n}.$
(8)

From (2), we have

$\displaystyle\mu_{0}(c)=\sum\limits_{x=0}^{\infty}\rho\big{(}x;\vec{\alpha},\vec{\beta},c\big{)}={}_{p}F_{q}\left[\begin{matrix}\vec{\alpha}\\\
\vec{\beta}\end{matrix};c\right],$ (9)

where ${}_{p}F_{q}$ is the generalized hypergeometric function [29, (16.2.1)].
Depending on the values of $p$ and $q$, we have to consider three different
cases:

  1. 1.

If $p<q+1$, $\mu_{0}(c)$ is an entire function of $c$.

  2. 2.

If $p=q+1$, $\mu_{0}(c)$ is analytic inside the unit disk $|c|<1$.

  3. 3.

If $p>q+1$, the series (9) diverges for $c\neq 0$, unless one or more of the
top parameters $\alpha_{i}$ is a negative integer. If we take $\alpha_{1}=-N$,
with $N\in\mathbb{N}$, then $\mu_{0}(c)$ becomes a polynomial of degree $N$.

Using the formula [29, (16.3.1)]

$\displaystyle\frac{d^{n}}{dc^{n}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{a}\\\
\vec{b}\end{matrix};c\right]=\frac{(\vec{a})_{n}}{(\vec{b})_{n}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{a}+n\\\
\vec{b}+n\end{matrix};c\right],$

in (7), we have

$\displaystyle\mu_{n}(c)=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}c^{k}\frac{(\vec{\alpha})_{k}}{(\vec{\beta}+1)_{k}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{\alpha}+k\\\
\vec{\beta}+1+k\end{matrix};c\right].$ (10)

Although (10) seems to give an explicit formula for the moments, this type of
sums (to our knowledge) can’t be evaluated in closed form.

An alternative is to consider _generalized moments_ , defined by [2]

$\displaystyle\nu_{n}=\mathcal{L}(\varphi_{n}),$

where $\\{\varphi_{n}\\}\subset\mathbb{C}[x]$, with $\deg(\varphi_{n})=n$.
Choosing444As suggested by one of the anonymous referees.

$\displaystyle\varphi_{n}(x)=(x-n+1)_{n},$

we get

$\displaystyle\nu_{n}(c)=\sum\limits_{x=0}^{\infty}(x-n+1)_{n}\frac{(\vec{\alpha})_{x}}{(\vec{\beta}+1)_{x}}\frac{c^{x}}{x!}=\frac{(\vec{\alpha})_{k}}{(\vec{\beta}+1)_{k}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{\alpha}+k\\\
\vec{\beta}+1+k\end{matrix};c\right].$

Since [29, (26.8.10)]

$\displaystyle
x^{n}=\sum\limits_{k=1}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}(x-k+1)_{k},$ (11)

we have

$\displaystyle\mu_{n}(c)=\mathcal{L}\big{(}x^{n}\big{)}=\sum\limits_{k=1}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}\mathcal{L}(\varphi_{k})=\sum\limits_{k=1}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}\nu_{k}(c),$

and we recover (10).

In a series of papers [7, 8, 10, 13, 14, 16], we studied polynomial solutions
of differential-difference equations of the form

$\displaystyle P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x),\qquad
n\geq 0,$ (12)

where $P_{0}(x)=1$, and $A_{n}(x)$, $B_{n}(x)$ are polynomials of degree at
most $2$ and $1$ respectively. In this article, we consider some extensions of
(12) to the multidimensional case, with $P_{n}(x)$ replaced by a vector
$\vec{P}_{n}(x)$, and $B_{n}(x)$ replaced by a matrix $\mathbf{B}_{n}(x)$.

In [19] and [20], it was shown that some families of orthogonal polynomials
can be represented as moments of probability measures. In this work, we do the
opposite and express the moments $\mu_{n}(c)$ in terms of some polynomials
$\vec{P}_{n}(c)$.

The paper is organized as follows: in Section 2, we derive a differential-
difference equation for the polynomials $\vec{P}_{n}(c)$ associated to the
moments $\mu_{n}(c)$. We also find formulas for the exponential generating
functions and Stieltjes transforms of $\mu_{n}(c)$ and
$\vec{P}_{n}\left(c\right)$. In Section 3, we apply our results to most of the
families of discrete semiclassical orthogonal polynomials of class 1 studied
in [15], except for limiting and $c=1$ cases. Finally, in Section 4, we
outline some conclusions and possible future directions.

##  2 Main results

The function $\mu_{0}(c)$ satisfies the differential equation [29, (16.8.3)]

$\displaystyle\big{[}\vartheta(\vartheta+\beta_{1})\cdots(\vartheta+\beta_{q})-c(\vartheta+\alpha_{1})\cdots(\vartheta+\alpha_{p})\big{]}\mu_{0}=0,$
(13)

where $\vartheta$ was defined in (6). We can rewrite the ODE (13) as

$\displaystyle\vartheta^{q+1}\mu_{0}=\sum\limits_{k=0}^{q}\sigma_{k}(c)\vartheta^{k}\mu_{0},\qquad
q>p-1,$
$\displaystyle(1-c)\vartheta^{q+1}\mu_{0}=\sum\limits_{k=0}^{q}\sigma_{k}(c)\vartheta^{k}\mu_{0},\qquad
q=p-1,$ (14) $\displaystyle
c\vartheta^{p}\mu_{0}=\sum\limits_{k=0}^{p-1}\sigma_{k}(c)\vartheta^{k}\mu_{0},\qquad
q<p-1,$

where the coefficients $\sigma_{k}(c)$ are linear functions of $c$.

Introducing the quantities

$\displaystyle\vec{\mu}(c)=\begin{bmatrix}\mu_{0}(c)\\\ \mu_{1}(c)\\\
\vdots\\\ \mu_{\xi}(c)\end{bmatrix},\qquad\xi=\max\\{p-1,q\\},$

and

$\displaystyle(\lambda,\tau)=\begin{cases}(1,0),&q+1>p,\\\ (1,-1),&q+1=p,\\\
(0,1),&q+1<p,\end{cases}$

we can rewrite (14) as

$\displaystyle(\lambda+c\tau)\mu_{\xi+1}=\sum\limits_{k=0}^{\xi}\sigma_{k}(c)\mu_{k}.$

If we define the $(\xi+1)\times(\xi+1)$ matrix $\mathbf{M}(c)$ by

$\displaystyle\mathbf{M}_{i,j}=\begin{cases}\sigma_{j},&i=\xi+1,\quad 0\leq
j\leq\xi,\\\ \lambda+c\tau,&j=i+1,\quad 0\leq i\leq\xi-1,\\\
0,&\text{otherwise},\end{cases}$

we get

$\displaystyle\mathbf{M}\,\vec{\mu}=(\lambda+c\tau)\vartheta\vec{\mu}.$ (15)

We can now state our main result.

######  Proposition 2.1.

Let the $(\xi+1)$-vector polynomials $\vec{P}_{n}(c)$ be defined by

$\displaystyle\vec{P}_{0}(c)=\begin{bmatrix}1\\\ 0\\\ \vdots\\\
0\end{bmatrix}$ (16)

and

$\displaystyle\vec{P}_{n+1}=c(\lambda+c\tau)\frac{d\vec{P}_{n}}{dc}+\big{(}\mathbf{M}^{T}-n\tau
c\mathbf{I}\big{)}\vec{P}_{n},\qquad n=0,1,\ldots,$ (17)

where $\mathbf{I}$ is the $(\xi+1)\times(\xi+1)$ identity matrix. Then,

$\displaystyle\mu_{n}(c)=(\lambda+c\tau)^{-n}\vec{P}_{n}(c)\cdot\vec{\mu}(c),\qquad
n=0,1,\ldots.$ (18)

######  Proof 2.2.

Using (15) and (17), we have

$\displaystyle\vec{P}_{n+1}\cdot\vec{\mu}=(\lambda+c\tau)\vartheta\vec{P}_{n}\cdot\vec{\mu}+\big{(}\mathbf{M}^{T}-n\tau
c\mathbf{I}\big{)}\vec{P}_{n}\cdot\vec{\mu}$
$\displaystyle\hphantom{\vec{P}_{n+1}\cdot\vec{\mu}}{}=(\lambda+c\tau)\vartheta\vec{P}_{n}\cdot\vec{\mu}+\vec{P}_{n}\cdot(\mathbf{M}-n\tau
c\mathbf{I})\vec{\mu}$
$\displaystyle\hphantom{\vec{P}_{n+1}\cdot\vec{\mu}}{}=(\lambda+c\tau)\vartheta\vec{P}_{n}\cdot\vec{\mu}+\vec{P}_{n}\cdot(\lambda+c\tau)\vartheta\vec{\mu}-n\tau
c\vec{P}_{n}\cdot\vec{\mu}.$

Multiplying by $(\lambda+c\tau)^{-n-1}$ we get

$\displaystyle(\lambda+c\tau)^{-n-1}\vec{P}_{n+1}\cdot\vec{\mu}=(\lambda+c\tau)^{-n}\vartheta\vec{P}_{n}\cdot\vec{\mu}+\vec{P}_{n}\cdot(\lambda+c\tau)^{-n}\vartheta\vec{\mu}-n\tau
c(\lambda+c\tau)^{-n-1}\vec{P}_{n}\cdot\vec{\mu}$
$\displaystyle\hphantom{(\lambda+c\tau)^{-n-1}\vec{P}_{n+1}\cdot\vec{\mu}}{}=\vartheta\big{[}(\lambda+c\tau)^{-n}\vec{P}_{n}\cdot\vec{\mu}\big{]}.$

Thus, the sequence

$\displaystyle r_{n}=(\lambda+c\tau)^{-n}\vec{P}_{n}\cdot\vec{\mu}$

satisfies the recurrence $r_{n+1}=\vartheta r_{n}$ with initial condition

$\displaystyle r_{0}=\vec{P}_{0}\cdot\vec{\mu}=\mu_{0}.$

From (5), we conclude that $r_{n}=\mu_{n}$.

###  2.1 Generating functions

Let’s consider the exponential generating function for the moments, defined by
the formal power series

$\displaystyle
G_{\mu}(c,w)=\sum\limits_{n=0}^{\infty}\mu_{n}(c)\frac{w^{n}}{n!}.$

Given that

$\displaystyle\mu_{n+1}=c\mu_{n}^{\prime},$

we have [33]

$\displaystyle\frac{\partial G_{\mu}}{\partial w}=c\frac{\partial
G_{\mu}}{\partial c},$

with general solution

$\displaystyle G_{\mu}(c,w)=H\big{(}ce^{w}\big{)}$

for some function $H(w)$. But since

$\displaystyle G_{\mu}(c,0)=\mu_{0}(c),$

we conclude that

$\displaystyle G_{\mu}(c,w)=\mu_{0}\big{(}ce^{w}\big{)}.$ (19)

Using (18), we see that the exponential generating function for the
polynomials $\vec{P}_{n}(c)$

$\displaystyle\vec{G}_{P}(c,w)=\sum\limits_{n=0}^{\infty}\vec{P}_{n}(c)\frac{w^{n}}{n!}$

satisfies

$\displaystyle G_{\mu}(c,(\lambda+\tau
c)w)=\vec{G}_{P}(c,w)\cdot\vec{\mu}(c),$

or, using (19),

$\displaystyle\vec{G}_{P}(c,w)\cdot\vec{\mu}(c)=\mu_{0}\big{(}ce^{(\lambda+\tau
c)w}\big{)}.$ (20)

###  2.2 Stieltjes transform

A different type of generating function for the moments that is very important
in the theory of orthogonal polynomials is the Stieltjes transform (or $Z$
transform), that can be defined by the formal Laurent series

$\displaystyle
S_{\mu}(c,z)=\sum\limits_{n=0}^{\infty}\frac{\mu_{n}(c)}{z^{n+1}}.$

We have

$\displaystyle
S_{\mu}(c,z)=\sum\limits_{n=0}^{\infty}\frac{\mu_{n}(c)}{z^{n+1}}=\sum\limits_{n=-1}^{\infty}\frac{\mu_{n+1}(c)}{z^{n+2}}=\frac{1}{z}\sum\limits_{n=-1}^{\infty}\frac{\mu_{n+1}(c)}{z^{n+1}}=\frac{1}{z}\left[\mu_{0}(c)+\sum\limits_{n=0}^{\infty}\frac{\mu_{n+1}(c)}{z^{n+1}}\right].$

Hence,

$\displaystyle\sum\limits_{n=0}^{\infty}\frac{\mu_{n+1}(c)}{z^{n+1}}=zS_{\mu}(c,z)-\mu_{0}(c)$
(21)

and using (5), we get

$\displaystyle c\frac{\partial S_{\mu}}{\partial c}=zS_{\mu}-\mu_{0}(c).$ (22)

From (4), we have

$\displaystyle S_{\mu}(0,z)=\frac{1}{z}$

and solving (22) we obtain

$\displaystyle
S_{\mu}(c,z)=-c^{z}\int_{0}^{c}\frac{\mu_{0}(x)}{x^{z+1}}dx=-\int_{0}^{1}\frac{\mu_{0}(ct)}{t^{z+1}}dt.$

Using the recurrence relation for the Gamma function, we can write

$\displaystyle-\int_{0}^{1}\frac{\mu_{0}(ct)}{t^{z+1}}dt=\frac{1}{z}\frac{\Gamma(1-z)}{\Gamma(1)\Gamma(-z)}\int_{0}^{1}\frac{\mu_{0}(ct)}{t^{z+1}}dt,$

and therefore

$\displaystyle
S_{\mu}(c,z)=\frac{1}{z}\,{}_{p+1}F_{q+1}\left[\begin{matrix}-z,\
\vec{\alpha}\\\ 1-z,\ \vec{\beta}+1\end{matrix};c\right],$ (23)

where we have used the integral representation [29, (16.5.2)]

$\displaystyle{}_{p+1}F_{q+1}\left[\begin{matrix}\alpha_{0},\ \vec{\alpha}\\\
\beta_{0},\
\vec{\beta}\end{matrix};c\right]=\frac{\Gamma(\beta_{0})}{\Gamma(\alpha_{0})\Gamma(\beta_{0}-\alpha_{0})}\int_{0}^{1}t^{\alpha_{0}-1}(1-t)^{\beta_{0}-\alpha_{0}-1}\,{}_{p}F_{q}\left[\begin{matrix}\vec{\alpha}\\\
\vec{\beta}\end{matrix};ct\right]dt.$

We derived (23) in [12] using a different approach.

If we define the Stieltjes transform of $\vec{P}_{n}(c)$ by

$\displaystyle\vec{S}_{P}(c,z)=\sum\limits_{n=0}^{\infty}\frac{\vec{P}_{n}(c)}{z^{n+1}},$

then it follows from (18) that

$\displaystyle\frac{1}{\lambda+\tau c}S_{\mu}\left(c,\frac{z}{\lambda+\tau
c}\right)=\vec{S}_{P}(c,z)\cdot\vec{\mu}(c),$

or

$\displaystyle\vec{S}_{P}(c,z)\cdot\vec{\mu}(c)=\frac{1}{z}\,{}_{p+1}F_{q+1}\left[\begin{matrix}-\frac{z}{\lambda+\tau
c},\ \vec{\alpha}\\\ 1-\frac{z}{\lambda+\tau c},\
\vec{\beta}+1\end{matrix};c\right].$ (24)

##  3 Examples

In [15] we studied all families of semiclassical polynomials of class $s\leq
1$ orthogonal with respect to (1). When $s=0$, we have three canonical cases
(the _discrete classical polynomials_):

$\deg(\eta)$ | $\deg(\phi)$ |
---|---|---
$0$ | $1$ | Charlier
$1$ | $1$ | Meixner
$2$ | $2$ | Hahn

where $\phi(x)$ and $\eta(x)$ were defined in (3). These polynomials are
associated with the Poisson, Pascal, and hypergeometric probability
distributions, and results about their moments have appeared in many places
before (see [21], for instance).

In [28], the authors used (11) and inversion formulas of the form

$\displaystyle(x-n+1)_{n}=\sum\limits_{k=0}^{n}c_{n,k}Q_{k}(x),$

to derive expressions for the moments and exponential generating functions of
the discrete classical polynomials. Stieltjes’ transforms were not considered.

When $s=1$, we obtained five cases:

$\deg(\eta)$ | $\deg(\phi)$ |
---|---|---
$0$ | $2$ | generalized Charlier
$1$ | $2$ | generalized Meixner
$2$ | $1$ | generalized Krawtchouk
$2$ | $2$ | generalized Hahn of type I
$3$ | $3$ | generalized Hahn of type II

The moments of these polynomials have not (to our knowledge) been studied
before.

###  3.1 Charlier

The Charlier polynomials [29, (18.20.8)]

$\displaystyle C_{n}(x;c)={}_{2}F_{0}\left(\begin{matrix}-n,-x\\\
-\end{matrix};-\frac{1}{c}\right)$

are orthogonal with respect to the weight function [6]

$\displaystyle\rho_{C}(x)=\frac{c^{x}}{x!},\qquad c>0.$

In this case, we have

$\displaystyle\mu_{0}(c)={}_{0}F_{0}\left[\begin{matrix}-\\\
-\end{matrix};c\right]=e^{c},$

and hence

$\displaystyle\xi=0,\qquad(\lambda,\tau)=(1,0).$

From (13) we see that $\mu_{0}(c)$ satisfies the ODE

$\displaystyle(\vartheta-c)\mu_{0}=0,$

which implies $\mu_{1}=c\mu_{0}$. Thus, $\sigma_{0}(c)=c$, and Proposition 2.1
gives

$\displaystyle\mu_{n}(c)=P_{n}(c)\mu_{0}(c),$

with $P_{n}(c)$ defined by $P_{0}(c)=1$ and

$\displaystyle P_{n+1}=c\frac{dP_{n}}{dc}+cP_{n}.$ (25)

The polynomials satisfying (25) are known as Bell (or Touchard, or
exponential) polynomials. It is well known that they have the explicit
representation [11]

$\displaystyle
P_{n}(c)=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}c^{k},$ (26)

and therefore (see also [28, equation (44)])

$\displaystyle\mu_{n}(c)=e^{c}\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}c^{k},$

in agreement with (10).

Using (19) and (20), we see that the generating functions of $\mu_{n}(c)$ and
$P_{n}(c)$ are given by (see also [28, equation (50)])

$\displaystyle G_{\mu}(c,w)=\mu_{0}\big{(}ce^{w}\big{)}=e^{ce^{w}},$

and

$\displaystyle G_{P}(c,w)=\frac{e^{ce^{w}}}{e^{c}}=e^{c(e^{w}-1)}.$

###  3.2 Meixner

The Meixner polynomials [29, (18.20.7)]

$\displaystyle M_{n}(x;\alpha,c)={}_{2}F_{1}\left(\begin{matrix}-n,-x\\\
\alpha\end{matrix};1-\frac{1}{c}\right)$

are orthogonal with respect to the weight function [29, (18.19)]

$\displaystyle\rho_{M}(x;\alpha,c)=(\alpha)_{x}\frac{c^{x}}{x!},\qquad
0<c<1,\qquad\alpha>0.$ (27)

In this case, we have

$\displaystyle\mu_{0}^{M}(c;\alpha)={}_{1}F_{0}\left[\begin{matrix}\alpha\\\
-\end{matrix};c\right]=(1-c)^{-\alpha},$ (28)

and hence

$\displaystyle\xi=0,\qquad(\lambda,\tau)=(1,-1).$

From (13) we see that $\mu_{0}^{M}(c;\alpha)$ satisfies the ODE

$\displaystyle[\vartheta-c(\vartheta+\alpha)]\mu_{0}^{M}=0,$

which implies

$\displaystyle(1-c)\mu_{1}^{M}=\alpha c\mu_{0}^{M}.$

Thus, $\sigma_{0}(c)=\alpha c$, and Proposition 2.1 gives

$\displaystyle\mu_{n}^{M}(c;\alpha)=(1-c)^{-n}P_{n}^{M}(c)\mu_{0}^{M}(c;\alpha),$
(29)

with $P_{n}^{M}(c)$ defined by $P_{0}^{M}(c)=1$ and (see also [27, equation
(5.68)])

$\displaystyle P_{n+1}^{M}=c(1-c)\frac{dP_{n}^{M}}{dc}+(\alpha+n)cP_{n}^{M}.$
(30)

From (10), we have (see also [28, equation (43)])

$\displaystyle\mu_{n}^{M}(c;\alpha)=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}c^{k}(\alpha)_{k}(1-c)^{-(\alpha+k)},$

and therefore

$\displaystyle
P_{n}^{M}(c)=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}(\alpha)_{k}c^{k}(1-c)^{n-k}.$
(31)

Using (19) and (20), we see that the generating functions of
$\mu_{n}^{M}(c;\alpha)$ and $P_{n}^{M}(c)$ are given by (see also [28,
equation (49)])

$\displaystyle
G_{\mu}(c,w)=\mu_{0}\big{(}ce^{w}\big{)}=\big{(}1-ce^{w}\big{)}^{-\alpha},$

and

$\displaystyle
G_{P}(c,w)=\frac{\big{[}1-ce^{(1-c)w}\big{]}^{-\alpha}}{(1-c)^{-\alpha}}=\left[\frac{1-ce^{(1-c)w}}{1-c}\right]^{-\alpha}.$

####  3.2.1 Krawtchouk polynomials

The Krawtchouk polynomials [29, (18.20.6)]

$\displaystyle K_{n}(x;\mathfrak{p,}N)={}_{2}F_{1}\left(\begin{matrix}-n,-x\\\
-N\end{matrix};\frac{1}{\mathfrak{p}}\right),$

are orthogonal with respect to the weight function [9]

$\displaystyle\rho_{K}(x;\mathfrak{p,}N)=\binom{N}{x}\mathfrak{p}^{x}(1-\mathfrak{p})^{N-x},\qquad
N\in\mathbb{N},\qquad 0<\mathfrak{p}<1.$

It is related to the weight function of the Meixner polynomials (27) by

$\displaystyle\rho_{K}(x;\mathfrak{p,}N)=(1-\mathfrak{p})^{N}\frac{(-N)_{x}}{x!}\left(\frac{\mathfrak{p}}{\mathfrak{p}-1}\right)^{x}=(1-\mathfrak{p})^{N}\rho_{M}\left(x;-N,\frac{\mathfrak{p}}{\mathfrak{p}-1}\right).$
(32)

Using (32) in (28), we obtain

$\displaystyle\mu_{0}^{K}(\mathfrak{p})=(1-\mathfrak{p})^{N}\mu_{0}^{M}\left(-N,\frac{\mathfrak{p}}{\mathfrak{p}-1}\right)=1.$

and using (29) and (31), we get
$\mu_{n}^{K}(\mathfrak{p})=P_{n}^{K}(\mathfrak{p})$, with (see also [28,
equation (42)])

$\displaystyle
P_{n}^{K}(\mathfrak{p})=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}\binom{N}{k}k!\mathfrak{p}^{k}.$

From (30) and (32), we have

$\displaystyle
P_{n+1}^{K}=\mathfrak{p}(1-\mathfrak{p})\frac{dP_{n}^{K}}{d\mathfrak{p}}+N\mathfrak{p}P_{n}^{K}.$

To obtain the generating function of $P_{n}^{K}(\mathfrak{p})$, we can use
[29, (26.8.12)]

$\displaystyle\sum\limits_{n=0}^{\infty}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}\frac{w^{n}}{n!}=\frac{(e^{w}-1)^{k}}{k!},$
(33)

and we get (see also [28, equation (48)])

$\displaystyle
G_{P}(\mathfrak{p},w)=\big{[}1+\big{(}e^{w}-1\big{)}\mathfrak{p}\big{]}^{N}.$

###  3.3 Generalized Charlier

These polynomials are orthogonal with respect to the weight function [17]

$\displaystyle\rho(x;\beta,c)=\frac{1}{(\beta+1)_{x}}\frac{c^{x}}{x!},\qquad
c>0,\qquad\beta>-1.$

In this case, we have

$\displaystyle\mu_{0}(c)={}_{0}F_{1}\left[\begin{matrix}-\\\
\beta+1\end{matrix};c\right]=c^{-\frac{\beta}{2}}\Gamma(\beta+1)I_{\beta}\big{(}2\sqrt{c}\big{)},$

where $I_{\beta}(z)$ is the modified Bessel function of the first kind [29,
(10.39.9)], and hence

$\displaystyle\xi=1,\qquad(\lambda,\tau)=(1,0).$

From (13) we see that $\mu_{0}(c)$ satisfies the ODE

$\displaystyle\vartheta(\vartheta+\beta)\mu_{0}-c\mu_{0}=0,$

which implies

$\displaystyle\mu_{2}=c\mu_{0}-\beta\mu_{1},$

and therefore

$\displaystyle\sigma_{0}(c)=c,\qquad\sigma_{1}(c)=-\beta.$

Proposition 2.1 gives

$\displaystyle\mu_{n}(c)=\vec{P}_{n}(c)\cdot\vec{\mu}(c),$

with $\vec{P}_{n}(c)$ defined by (16) and

$\displaystyle\vec{P}_{n+1}=c\frac{d\vec{P}_{n}}{dc}+\begin{bmatrix}0&c\\\
1&-\beta\end{bmatrix}\vec{P}_{n}.$ (34)

To obtain the Stieltjes transform of $\vec{P}_{n}(c)$, let’s write

$\displaystyle\vec{P}_{n}(c)=\left[\begin{matrix}Q_{n}(c)\\\
R_{n}(c)\end{matrix}\right],\qquad\vec{S}_{P}(c,z)=\left[\begin{matrix}U(c,z)\\\
V(c,z)\end{matrix}\right].$

From (34), we have

$\displaystyle Q_{n+1}=cQ_{n}^{\prime}+cR_{n},\qquad
R_{n+1}=cR_{n}^{\prime}-\beta R_{n}+Q_{n},$

with $Q_{0}=1$, $R_{0}=0$. Using (21), we get

$\displaystyle zU-1=c\frac{\partial U}{\partial c}+cV,\qquad
zV=c\frac{\partial V}{\partial c}-\beta V+U.$ (35)

The solution of the system (35) is given by

$\displaystyle U(c,z)=\frac{1}{z}\,{}_{1}F_{2}\left[\begin{matrix}1\\\ 1-z,\
-\beta-z\end{matrix};c\right],\qquad
V(c,z)=\frac{1}{z(z+\beta)}\,{}_{1}F_{2}\left[\begin{matrix}1\\\ 1-z,\
1-\beta-z\end{matrix};c\right].$

In this case

$\displaystyle\mu_{0}(c)={}_{0}F_{1}\left[\begin{matrix}-\\\
\beta+1\end{matrix};c\right],\qquad\mu_{1}(c)=\frac{c}{\beta+1}\,{}_{0}F_{1}\left[\begin{matrix}-\\\
\beta+2\end{matrix};c\right],$

and since

$\displaystyle\frac{1}{z}\,{}_{1}F_{2}\left[\begin{matrix}1\\\ 1-z,\
-\beta-z\end{matrix};c\right]\,{}_{0}F_{1}\left[\begin{matrix}-\\\
\beta+1\end{matrix};c\right]$
$\displaystyle\qquad{}+\frac{1}{z(z+\beta)}\,{}_{1}F_{2}\left[\begin{matrix}1\\\
1-z,\
1-\beta-z\end{matrix};c\right]\frac{c}{\beta+1}\,{}_{0}F_{1}\left[\begin{matrix}-\\\
\beta+2\end{matrix};c\right]=\frac{1}{z}\,{}_{1}F_{2}\left[\begin{matrix}-z\\\
1-z,\ \beta+1\end{matrix};c\right],$

we see that (24) is satisfied.

###  3.4 Generalized Meixner

These polynomials are orthogonal with respect to the weight function [3]

$\displaystyle\rho(x;\alpha,\beta,c)=\frac{(\alpha)_{x}}{(\beta+1)_{x}}\frac{c^{x}}{x!},\qquad
c,\alpha>0,\qquad\beta>-1.$

In this case, we have

$\displaystyle\mu_{0}(c)={}_{1}F_{1}\left[\begin{matrix}\alpha\\\
\beta+1\end{matrix};c\right]=M(\alpha,\beta+1,c),$

where $M(a,b,c)$ is Kummer’s (confluent hypergeometric) function [29,
(13.2.2)], and hence

$\displaystyle\xi=1,\qquad(\lambda,\tau)=(1,0).$

From (13) we see that $\mu_{0}(c)$ satisfies the ODE

$\displaystyle\vartheta(\vartheta+\beta)\mu_{0}-c(\vartheta+\alpha)\mu_{0}=0,$

which implies

$\displaystyle\mu_{2}=c\alpha\mu_{0}+(c-\beta)\mu_{1},$

and therefore

$\displaystyle\sigma_{0}(c)=\alpha c,\qquad\sigma_{1}(c)=c-\beta.$

Proposition 2.1 gives

$\displaystyle\mu_{n}(c)=\vec{P}_{n}(c)\cdot\vec{\mu}(c),$

with $\vec{P}_{n}(c)$ defined by (16) and

$\displaystyle\vec{P}_{n+1}=c\frac{d\vec{P}_{n}}{dc}+\begin{bmatrix}0&\alpha
c\\\ 1&c-\beta\end{bmatrix}\vec{P}_{n}.$

To obtain the Stieltjes transform of $\vec{P}_{n}(c)$, let’s write

$\displaystyle\vec{P}_{n}(c)=\left[\begin{matrix}Q_{n}(c)\\\
R_{n}(c)\end{matrix}\right],\qquad\vec{S}_{P}(c,z)=\left[\begin{matrix}U(c,z)\\\
V(c,z)\end{matrix}\right].$

From (34), we have

$\displaystyle Q_{n+1}=cQ_{n}^{\prime}+\alpha cR_{n},\qquad
R_{n+1}=cR_{n}^{\prime}+(c-\beta)R_{n}+Q_{n},$

with $Q_{0}=1$, $R_{0}=0$. Using (21), we get

$\displaystyle zU-1=c\frac{\partial U}{\partial c}+\alpha cV,\qquad
zV=c\frac{\partial V}{\partial c}+(c-\beta)V+U.$ (36)

If we represent the functions $U$, $V$ as

$\displaystyle U(c,z)=\sum\limits_{n=0}^{\infty}u_{n}(z)c^{n},\qquad
V(c,z)=\sum\limits_{n=0}^{\infty}v_{n}(z)c^{n},$

then (36) gives

$\displaystyle zu_{n}-\delta_{n,0}=nu_{n}+\alpha v_{n-1},\qquad
zv_{n}=nv_{n}+v_{n-1}-\beta v_{n}+u_{n}.$ (37)

We have (assuming that $u_{n}=v_{n}=0$ for $n<0)$

$\displaystyle u_{0}=\frac{1}{z},\qquad v_{0}=\frac{1}{z(z+\beta)},$

and

$\displaystyle u_{n}=(z+\beta-n)v_{n}-v_{n-1}.$ (38)

Using (38) in (37), we obtain

$\displaystyle v_{n}=\frac{z+\alpha-n}{(z-n)(z+\beta-n)}v_{n-1},\qquad
n=1,2,\ldots,$

and therefore

$\displaystyle
v_{n}=\frac{(-1)^{n}}{z(z+\beta)}\frac{(1-\alpha-z)_{n}}{(1-z)_{n}(1-\beta-z)_{n}},\qquad
n=0,1,\ldots.$

From (38), it follows that

$\displaystyle
u_{n}=\alpha\frac{(-1)^{n}}{z(z+\alpha)}\frac{(-\alpha-z)_{n}}{(1-z)_{n}(-\beta-z)_{n}},\qquad
n=1,2,\ldots.$

Thus,

$\displaystyle
U(c,z)=\frac{1}{z}+\frac{\alpha}{z(z+\alpha)}\sum\limits_{n=1}^{\infty}\frac{(-\alpha-z)_{n}}{(1-z)_{n}(-\beta-z)_{n}}(-c)^{n}$
$\displaystyle\hphantom{U(c,z)}{}=\frac{1}{z}-\frac{\alpha}{z(z+\alpha)}+\frac{\alpha}{z(z+\alpha)}\sum\limits_{n=0}^{\infty}\frac{(-\alpha-z)_{n}}{(1-z)_{n}(-\beta-z)_{n}}(-c)^{n}$
$\displaystyle\hphantom{U(c,z)}{}=\frac{1}{z+\alpha}+\frac{\alpha}{z(z+\alpha)}\,{}_{2}F_{2}\left[\begin{matrix}1,-\alpha-z\\\
1-z,\ -\beta-z\end{matrix};-c\right],$

and

$\displaystyle V(c,z)=\frac{1}{z(z+\beta)}\,{}_{2}F_{2}\left[\begin{matrix}1,\
1-\alpha-z\\\ 1-z,\ 1-\beta-z\end{matrix};-c\right].$

In this case

$\displaystyle\mu_{0}(c)={}_{1}F_{1}\left[\begin{matrix}\alpha\\\
\beta+1\end{matrix};c\right],\qquad\mu_{1}(c)=\frac{\alpha
c}{\beta+1}\,{}_{1}F_{1}\left[\begin{matrix}\alpha+1\\\
\beta+2\end{matrix};c\right],$

and since

$\displaystyle\left(\frac{1}{z+\alpha}+\frac{\alpha}{z(z+\alpha)}\,{}_{2}F_{2}\left[\begin{matrix}1,-\alpha-z\\\
1-z,\
-\beta-z\end{matrix};-c\right]\right)\,{}_{1}F_{1}\left[\begin{matrix}\alpha\\\
\beta+1\end{matrix};c\right]$
$\displaystyle\\!\qquad{}+\frac{1}{z(z+\beta)}\,{}_{2}F_{2}\left[\begin{matrix}1,\
1-\alpha-z\\\ 1-z,\ 1-\beta-z\end{matrix};-c\right]\frac{\alpha
c}{\beta+1}\,{}_{1}F_{1}\left[\begin{matrix}\alpha+1\\\
\beta+2\end{matrix};c\right]=\frac{1}{z}\,{}_{2}F_{2}\left[\begin{matrix}-z,\
\alpha\\\ 1-z,\ \beta+1\end{matrix};c\right],$

we see that (24) is satisfied.

###  3.5 Generalized Krawtchouk

These polynomials are orthogonal with respect to the weight function [15]

$\displaystyle\rho(x;\alpha,N,c)=(\alpha)_{x}(-N)_{x}\frac{c^{x}}{x!},\qquad
c<0,\qquad\alpha>0,\qquad N\in\mathbb{N}.$

In this case, we have

$\displaystyle\mu_{0}(c)={}_{2}F_{0}\left[\begin{matrix}\alpha,\ -N\\\
-\end{matrix};c\right]=C_{N}\big{(}{-}\alpha;-c^{-1}\big{)},$

where $C_{N}(x;\mu)$ is the Charlier polynomial [6], and hence

$\displaystyle\xi=1,\qquad(\lambda,\tau)=(0,1).$

From (13) we see that $\mu_{0}(c)$ satisfies the ODE

$\displaystyle\vartheta\mu_{0}-c(\vartheta+\alpha)(\vartheta-N)\mu_{0}=0,$

which implies

$\displaystyle c\mu_{2}=\alpha Nc\mu_{0}+(Nc-\alpha c+1)\mu_{1},$

and therefore

$\displaystyle\sigma_{0}(c)=\alpha Nc,\qquad\sigma_{1}(c)=(N-\alpha)c+1.$

Proposition 2.1 gives

$\displaystyle\mu_{n}(c)=c^{-n}\vec{P}_{n}(c)\cdot\vec{\mu}(c),$

with $\vec{P}_{n}(c)$ defined by (16) and

$\displaystyle\vec{P}_{n+1}=c^{2}\frac{d\vec{P}_{n}}{dc}+\begin{bmatrix}-nc&\alpha
Nc\\\ c&\left(N-\alpha-n\right)c+1\end{bmatrix}\vec{P}_{n}(c).$

###  3.6 Generalized Hahn polynomials of type I

These polynomials are orthogonal with respect to the weight function [15]

$\displaystyle\rho(x;\alpha_{1},\alpha_{2},\beta;c)=\frac{(\alpha_{1})_{x}(\alpha_{2})_{x}}{(\beta+1)_{x}}\frac{c^{x}}{x!},$
(39)

where $0<c<1$, $\alpha_{1},\alpha_{2}>0$, $\beta>-1$. In this case, we have

$\displaystyle\mu_{0}(c)={}_{2}F_{1}\left[\begin{matrix}\alpha_{1},\
\alpha_{2}\\\ \beta+1\end{matrix};c\right],$

and hence

$\displaystyle\xi=1,\qquad(\lambda,\tau)=(1,-1).$

From (13) we see that $\mu_{0}(c)$ satisfies the ODE

$\displaystyle\vartheta(\vartheta+\beta)\mu_{0}-c(\vartheta+\alpha_{1})(\vartheta+\alpha_{2})\mu_{0}=0,$

which implies

$\displaystyle(1-c)\mu_{2}=\allowbreak\alpha_{1}\alpha_{2}c\mu_{0}+\big{[}(\alpha_{1}+\alpha_{2})c-\beta\big{]}\mu_{1},$

and therefore

$\displaystyle\sigma_{0}(c)=\alpha_{1}\alpha_{2}c,\qquad\sigma_{1}(c)=(\alpha_{1}+\alpha_{2})c-\beta.$

Proposition 2.1 gives

$\displaystyle\mu_{n}(c)=(1-c)^{-n}\vec{P}_{n}(c)\cdot\vec{\mu}(c),$

with $\vec{P}_{n}(c)$ defined by (16) and

$\displaystyle\vec{P}_{n+1}=c(1-c)\frac{d\vec{P}_{n}}{dc}+\begin{bmatrix}nc&\alpha_{1}\alpha_{2}(1-c)c\\\
1-c&(\alpha_{1}+\alpha_{2}+n)c-\beta\end{bmatrix}\vec{P}_{n}.$

###  3.7 Hahn polynomials

The Hahn polynomials [29, (18.20.5)]

$\displaystyle
H_{n}(x;\alpha,\beta,N)={}_{3}F_{2}\left(\begin{matrix}-n,-x,n+\alpha+\beta+1\\\
-N,\alpha+1\end{matrix};1\right)$

are orthogonal with respect to the weight function [29, (18.19)]

$\displaystyle\rho_{H}(x;\alpha,\beta,N)=\frac{(\alpha+1)_{x}}{x!}\frac{(\beta+1)_{N-x}}{(N-x)!},\qquad\alpha,\beta\notin[-N,-1],\qquad
N\in\mathbb{N},$

or

$\displaystyle\rho_{H}(x;\alpha,\beta,N)=\binom{\alpha+x}{x}\binom{\beta+N-x}{N-x}.$

The relation between $\rho_{H}(x;\alpha,\beta,N)$ and the weight function of
the generalized Hahn polynomials (39) is given by

$\displaystyle\rho_{H}(x;\alpha,\beta,N)=\frac{(\beta+1)_{N}}{N!}\rho(x;\alpha+1,-N,-N-\beta-1;1).$

From (10), we get (see also [28, equation (41)])

$\displaystyle\mu_{n}^{H}=\sum\limits_{k=0}^{n}\genfrac{\\{}{\\}}{0.0pt}{}{n}{k}(\alpha+1)_{k}\frac{(\alpha+\beta+2+k)_{N-k}}{(N-k)!},$

where we have used

$\displaystyle\mu_{0}^{H}=\frac{(\alpha+\beta+2)_{N}}{N!}.$

To obtain the generating function of $\mu_{n}^{H}$, we can use (33), and we
get

$\displaystyle
G_{\mu^{H}}(z)=\frac{(\alpha+\beta+2)_{N}}{N!}\,{}_{2}F_{1}\left[\begin{matrix}-N,\
\alpha+1\\\ \alpha+\beta+2\end{matrix};1-e^{z}\right].$

This generating function seems not to have been considered before.

######  Remark 3.1.

Except for the Bell polynomials (26), all the other families $\vec{P}_{n}(c)$
associated to the moments $\mu_{n}(c)$ seem to be new.

######  Remark 3.2.

Stieltjes transforms of the generalized Krawtchouk and generalized Hahn
polynomials of type I have been omitted because of the complexity of the
formulas.

##  4 Conclusion

We have developed a technique for computing the moments of weight functions of
hypergeometric type. We have shown that the moments are linear combinations of
the first $\xi+1$ moments with polynomial coefficients in the parameter $c$.
We have also constructed generating functions for both the moments and the
polynomials associated with them.

All the results in Sections 3.3–3.5 are new, and give efficient ways of
computing the moments of the discrete semiclassical polynomials of class $1$.
The same method can be used to find the moments of polynomials of class $s>1$.

In a previous work [16], we found the asymptotic zero distribution of
polynomial families satisfying first-order differential-recurrence relations
of the form (12). It would be interesting to know if our results could be
extended to include the polynomials $\vec{P}_{n}(c)$ studied in this paper.

### Acknowledgements

We want to thank the anonymous referees and the editors of this special issue
for providing us with extremely valuable suggestions that helped us to greatly
improve our first draft of the paper.

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