# A Characterization of Umbral Calculus Inspired by Fractional Sums

Tang Qian

###### Abstract

We will use analytic function theory and Fourier analysis to establish a
characterization for some classical umbral calculus, which will focus on the
generalization of the evaluation function. Although we cannot cover all the
umbral calculus people care about, the part about Bernoulli numbers can still
answer an open question about fractional sums raised by Müller and Schleicher
in 2005 [10] and synthesize some common analysis results. We will only develop
the results which are sufficient to serve the purpose of this article, but at
the same time, we will briefly mention some possible extensions.

##  1 Introduction

Umbral calculus is a basic technique in combinatorics, systematized by Gian-
Carlo Rota [13]. Generally speaking, for a given sequence
$(A_{n})_{n\in\mathbb{N}}$, it asks us to treat $A_{n}$ as $A^{n}$ to simplify
some combinatorics deduction and revealed some imperceptible results. [4]
gives an appropriate introduction about the applications, and [2] gives a
comprehensive survey. Consider Bernoulli numbers
$(\mathrm{B}_{n})_{n\in\mathbb{N}}$, we can write Faulhaber’s formula as

$\sum_{k=1}^{n}k^{m}=\frac{1}{m+1}\sum_{k=0}^{m}\binom{m+1}{k}\mathrm{B}_{k}n^{m+1-k}=\frac{(\mathrm{B}+n)^{m+1}-\mathrm{B}^{m+1}}{m+1}.$

Therefore for polynomial $P$, we have

$\sum_{k=1}^{n}P(k)=\int_{0}^{n}P(\mathrm{B}+k)\mathrm{d}k.$ (1.1)

Those results strongly hint that we can consider replacing the polynomial $P$
and the integer $n$ in (1.1) by function $f$ and complex number $z$. However,
it requires us to explain $f(\mathrm{B})$ and $\sum_{k=1}^{z}$ appropriately ,
and we need more motivation for doing this.

Fractional sums is a concept first formulated clearly by Müller and Schleicher
[10, 11, 12]. They proposed six natural axioms on $\sum_{k=a}^{b}$, for
appropriate function $f$ and complex number $a,b$, those axioms can derive the
value of $\sum_{k=a}^{b}f(k)$ uniquely. Their results can be reformulated as
below.

######  Definition 1.1 (Müller-Schleicher fractional summation).

Suppose $U\subseteq\mathbb{C}$ satisfied $U+1\subseteq U$ and $1\in U$, $f$ is
a complex value function on $U$.

If there is a polynomial sequence $(P_{n})_{n\in\mathbb{N}}$ satisfied

  1. 1.

The degree of the polynomial sequence $(P_{n})_{n\in\mathbb{N}}$ is uniformly
bounded;

  2. 2.

For all $y+1\in U$, the limits

$\lim\limits_{n\to\infty}\left(\sum_{k=1}^{n}(f(k+y)-f(k))-P_{n}(y)\right)=-Q(y)$

exists;

  3. 3.

For all $y+1\in U$,

$Q(y+1)-Q(y)=f(y+1),$

then we defined Müller-Schleicher fractional summation as

$(\mathrm{MS})\text{-}\sum_{k=x}^{y}f(k)\coloneqq Q(y)-Q(x-1),$ (1.2)

where $x,y+1\in U$.

For the well-definedness of Müller-Schleicher fractional summation(for short,
MS-fractional summation), we referred to [11]. Many classical infinite sum
identities can be reformulated in fractional summation and revealed a direct
proof [10].

In those reformulations, especially for the Gosper series, which Gosper
discussed and proved by the properties of Bessel function [5].

$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n+\frac{1}{2}}\frac{\sin\sqrt{b^{2}+\pi^{2}(n+1/2)^{2}}}{\sqrt{b^{2}+\pi^{2}(n+1/2)^{2}}}$
$\displaystyle=\frac{\pi\sin b}{2b}$ (1.3)
$\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos\sqrt{b^{2}+\pi^{2}n^{2}}$
$\displaystyle=\frac{\pi^{2}}{4}\left(\frac{\sin b}{b}-\frac{\cos
b}{3}\right).$ (1.4)

They noticed that [10, 11, 12] those series could be proved directly by an
interchange of a fractional sum and an infinite series, i.e.

$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n+\frac{1}{2}}\frac{\sin\sqrt{b^{2}+\pi^{2}(n+1/2)^{2}}}{\sqrt{b^{2}+\pi^{2}(n+1/2)^{2}}}$
$\displaystyle=(\mathrm{MS})\text{-}\sum_{n=1/4}^{-1/4}\frac{1}{2n}\frac{\sin\sqrt{b^{2}+(2\pi
n)^{2}}}{\sqrt{b^{2}+(2\pi n)^{2}}}$
$\displaystyle\overset{?}{=}(\mathrm{MS})\text{-}\frac{\sin
b}{2b}\sum_{n=1/4}^{-1/4}n^{-1},$

and

$\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n^{2}}\cos\sqrt{b^{2}+\pi^{2}n^{2}}$
$\displaystyle=(\mathrm{MS})\text{-}\frac{1}{4}\sum_{n=1}^{-1/2}\frac{1}{n^{2}}\cos\sqrt{b^{2}+(2\pi
n)^{2}}$
$\displaystyle\overset{?}{=}(\mathrm{MS})\text{-}\frac{1}{4}\left(\cos
b\sum_{n=1}^{-1/2}n^{-2}-\frac{2\pi^{2}\sin b}{b}\sum_{n=1}^{-1/2}1\right).$

Where

$\displaystyle\frac{1}{2n}\frac{\sin\sqrt{b^{2}+(2\pi
n)^{2}}}{\sqrt{b^{2}+(2\pi n)^{2}}}$ $\displaystyle=\frac{\sin
b}{2b}n^{-1}+a_{1}n+a_{3}n^{3}+a_{5}n^{5}+\cdots$
$\displaystyle\frac{1}{n^{2}}\cos\sqrt{b^{2}+(2\pi n)^{2}}$
$\displaystyle=\cos b\cdot n^{-2}-\frac{2\pi^{2}\sin
b}{b}+a_{2}n^{2}+a_{4}n^{4}+\cdots,$

and we have

$(\mathrm{MS})\text{-}\sum_{n=1/4}^{-1/4}n^{2k-1}=0\quad(\mathrm{MS})\text{-}\sum_{n=1}^{-1/2}n^{2k}=0$
$(\mathrm{MS})\text{-}\sum_{n=1/4}^{-1/4}n^{-1}=\pi\quad(\mathrm{MS})\text{-}\sum_{n=1}^{-1/2}n^{-2}=-\frac{\pi^{2}}{3}\quad(\mathrm{MS})\text{-}\sum_{n=1}^{-1/2}1=-\frac{1}{2},$

for all positive integer $k$. We have enough motivation to determine the
sufficient condition on function for this interchangeability. In [11] they
also concerned about the conditions that validated the following formula

$(\mathrm{MS})\text{-}\frac{\mathrm{d}}{\mathrm{d}x}\sum_{k=1}^{x}f(k)=c_{f}+(\mathrm{MS})\text{-}\sum_{k=1}^{x}\frac{\mathrm{d}}{\mathrm{d}k}f(k).$
(1.5)

Our goal is to justify those speculation.

###  1.1 Framework

Our main idea is generalized (1.1) to

$(\mathrm{MS})\text{-}\sum_{k=1}^{z}f(k)=F(\mathrm{B}+z)-F(\mathrm{B}),$ (1.6)

where $F$ is the primitive function of $f$, and transferred the difficulty of
interchangeability to the definition and properties of $F(\mathrm{B})$, i.e.
$\mathrm{eval}_{\mathrm{B}}(F)$. We achieved this in Theorem 9.

In the next section, we will discuss the evaluation function including a kind
of sequence $(A_{n})_{n\in\mathbb{N}}$, such as Bernoulli numbers, and
developed all the properties we need. This is achieved by the analytic
function theory and Fourier transform, i.e.

$f(A)\coloneqq\mathrm{eval}_{A}(f)\coloneqq\frac{1}{\sqrt{2\pi}}\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\hat{\mathcal{A}}(z)f(\mathrm{i}z)\mathrm{d}z,$
(1.7)

where $\mathcal{A}(z)$ is the generating function of sequence
$(A_{n})_{n\in\mathbb{N}}$. In fact, similar ideas can be founded in other
places. In [8, 9], the author treats $A_{n}$ as the $n$-th moment of some
signed measure $\mu$; in [14], the author also noticed connections with
Fourier analysis; in [6, 7], the author gives an extension of the umbral
calculus to certain classes of entire function. Our discussion will follow a
different way.

Section 3 will focus on the applications of section 2, Corollary 3.1, 3.2
includes the validity of the speculation mentioned above and determined
$c_{f}$. Finally, in Corollary 3.3, we proved and promoted some conclusions in
[5] as expected by Müller and Schleicher.

In Section 4 we introduced some possible developments.

##  2 The Analysis of Umbrae

The Fourier transform of $f\in L^{1}(\mathbb{R})$, which we take is

$\hat{f}(\xi)\coloneqq\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)\mathrm{e}^{-\mathrm{i}\xi
x}\mathrm{d}x,$

and it also induced the Fourier transform on slowly increasing distribution
space $\mathcal{S}^{\prime}(\mathbb{R})$. Sometimes we would interpret the
integral of the Fourier transform in the meaning of the summation method.

For our purpose, we will generalize the concept of the Fourier transform as
below.

######  Definition 2.1 (Fourier transform).

If $f$ is defined on $\mathbb{R}-\mathrm{i}t$, then we define the Fourier
transform of $f$ is

$\displaystyle\hat{f}(\xi-\mathrm{i}s)$
$\displaystyle\coloneqq\frac{1}{\sqrt{2\pi}}\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}f(z)\mathrm{e}^{-\mathrm{i}(\xi-\mathrm{i}s)z}\mathrm{d}z$
(2.1)
$\displaystyle=\frac{\mathrm{e}^{-(\xi-\mathrm{i}s)t}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{e}^{-sx}f(x-\mathrm{i}t)\mathrm{e}^{-\mathrm{i}\xi
x}\mathrm{d}x.$

It should be emphasized that $f$ could be exponential growth, and $\hat{f}$
could be undefined on $\mathbb{R}$.

######  Definition 2.2 (Umbrae).

If $\mathcal{A}$ is an exponential type analytic function defined on

$\Omega_{a,b}\coloneqq\\{x-\mathrm{i}t:x\in\mathbb{R},t\in(a,b),\\}$

i.e. there exists $s\in\mathbb{R}$ such that for every $t\in(a,b)$,

$\left|\mathcal{A}(x-\mathrm{i}t)\right|\leq C(t)\mathrm{e}^{s|x|},$

where $C$ is locally bounded, then

  1. 1.

we called $A=(\mathcal{A},\Omega_{a,b})$ is a umbrae;

  2. 2.

we called $\mathcal{A}$ is the generating function of $A$.

When $0\in\Omega_{a,b}$, the umbrae $A$ exactly corresponds to the sequence
$(\mathcal{A}^{(n)}(0))_{n\in\mathbb{N}}$. $s$ can be treated as dependent on
$t$, but we do not need such a detailed treatment here.

######  Lemma 2.1.

Suppose $f$ is an exponential type analytic function defined on
$\Omega_{a,b}$. $t_{1},t_{2}\in(a,b)$.
If $f$ is Lebesgue integrable on
$\mathbb{R}-\mathrm{i}t_{1},\mathbb{R}-\mathrm{i}t_{2}$, then

$\int_{-\infty-\mathrm{i}t_{1}}^{\infty-\mathrm{i}t_{1}}f(z)\mathrm{d}z=\int_{-\infty-\mathrm{i}t_{2}}^{\infty-\mathrm{i}t_{2}}f(z)\mathrm{d}z.$
(2.2)

###### Proof.

First notice that $\mathrm{e}^{-\varepsilon z^{2}}f(z)$ locally uniformly
tends to $0$ when $|z|\to\infty$, therefore

$\int_{-\infty-\mathrm{i}t_{1}}^{\infty-\mathrm{i}t_{1}}\mathrm{e}^{-\varepsilon
z^{2}}f(z)\mathrm{d}z=\int_{-\infty-\mathrm{i}t_{2}}^{\infty-\mathrm{i}t_{2}}\mathrm{e}^{-\varepsilon
z^{2}}f(z)\mathrm{d}z.$

By the dominated convergence theorem, only need to let $\varepsilon\to 0$. ∎

This lemma is also valid for larger function classes, but again the
exponential type condition is sufficient for our needs.

######  Lemma 2.2 ($L^{1}$-Phragmén–Lindelöf principle).

Suppose $f$ is an exponential type analytic function defined on
$\Omega_{a,b}$. $t_{1},t_{2}\in(a,b),t_{1}<t_{2}$.
If $f$ is Lebesgue integrable on
$\mathbb{R}-\mathrm{i}t_{1},\mathbb{R}-\mathrm{i}t_{2}$, then for every
$t\in[t_{1},t_{2}]$, $f$ is Lebesgue integrable on $\mathbb{R}-\mathrm{i}t$,
and $\ln\|f(\cdot-\mathrm{i}t)\|_{1}$ is a convex function for
$t\in[t_{1},t_{2}]$.

###### Proof.

See [3] p.479 Theorem 4. ∎

###  2.1 Basic definition and properties

We basically only care about a special class of umbrae.

######  Definition 2.3.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a umbrae.

  1. 1.

Denote

$\displaystyle\alpha$
$\displaystyle=\inf\\{s\in\mathbb{R}:\exists\text{locally bounded C}.\forall
x\geq 0.|f(x-\mathrm{i}t)|\leq C(t)\mathrm{e}^{sx}\\}$ $\displaystyle\beta$
$\displaystyle=\sup\\{s\in\mathbb{R}:\exists\text{locally bounded C}.\forall
x\leq 0.|f(x-\mathrm{i}t)|\leq C(t)\mathrm{e}^{sx}\\}.$

We called $\alpha$ is the positive index; $\beta$ is the negative index;
$(\alpha,\beta)$ is the index of the umbrae $A$;

  2. 2.

If $\alpha\geq\beta$, we called umbrae $A$ is singular;

  3. 3.

If $\alpha<\beta$, we called umbrae $A$ is regular, and the open interval
$(\alpha,\beta)$ is called the regular interval of $A$;

  4. 4.

We called the open interval $(a,b)$ is the dominating interval of $A$.

The reasons for choosing these terms and signs will become clearer in the
following discussion.

######  Definition 2.4 (Umbrae calculus).

Suppose $A_{i}=(\mathcal{A}_{i},\Omega_{a_{i},b_{i}})$ is a umbrae.

  1. 1.

For $r\in\mathbb{R}$, $rA\coloneqq(\mathcal{A}(rz),r^{-1}\Omega_{a,b})$;

  2. 2.

$A_{1}+A_{2}\coloneqq(\mathcal{A}_{1}\mathcal{A}_{2},\Omega_{a_{1},b_{1}}\cap\Omega_{a_{2},b_{2}})$,
$A_{1}-A_{2}\coloneqq A_{1}+(-1)A_{2}$;

  3. 3.

For $n\in\mathbb{N}$, $0\times A\coloneqq 0,(n+1)\times A\coloneqq(n\times
A)+A$;

  4. 4.

$A_{1}[+]A_{2}\coloneqq(\mathcal{A}_{1}+\mathcal{A}_{2},\Omega_{a_{1},b_{1}}\cap\Omega_{a_{2},b_{2}})$;

  5. 5.

$A_{1}[-]A_{2}\coloneqq(\mathcal{A}_{1}-\mathcal{A}_{2},\Omega_{a_{1},b_{1}}\cap\Omega_{a_{2},b_{2}})$.

The scalar multiplication and addition in the above definition come from the
classic umbral calculus. They do not satisfy the properties that these
operations should normally have, but this definition is convenient for
evaluation, and we will verify its well-definedness later.

######  Definition 2.5 (Special umbrae).

  1. 1.

For $c\in\mathbb{C}$, $(c)\coloneqq(\mathrm{e}^{cz},\mathbb{C})$, the index is
$(\mathrm{Re}c,\mathrm{Re}c)$;

  2. 2.

For $c\in\mathbb{C}$, $[c]\coloneqq(c,\mathbb{C})$, the index is
$\begin{cases*}(0,0),&$c\neq 0$\\\ (-\infty,\infty),&$c=0$\end{cases*}$

  3. 3.

$\mathrm{D}\coloneqq(z,\mathbb{C})$, the index is $(0,0)$;

  4. 4.

$\Delta\coloneqq(\mathrm{e}^{z}-1,\mathbb{C})$, the index is $(1,0)$;

  5. 5.

$\mathrm{B}\coloneqq(\frac{z\mathrm{e}^{z}}{\mathrm{e}^{z}-1},\Omega_{-2\pi,2\pi})$,
the index is $(0,1)$;

  6. 6.

$\mathrm{E}\coloneqq(\frac{2}{\mathrm{e}^{z}+\mathrm{e}^{-z}},\Omega_{-\frac{\pi}{2},\frac{\pi}{2}})$,
the index is $(-1,1)$.

In fact, we will identify the complex number $c$ to the umbrae $(c)$.

For regular umbrae $A=(\mathcal{A},\Omega_{a,b})$, we can use Definition 2.1
to calculate the Fourier transform of $\mathcal{A}$. By the knowledge of
analysis, it is not difficult to see that $\hat{\mathcal{A}}$ is well-defined,
and it also induced a regular umbrae.

######  Theorem 1 (Correspondence).

If $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae with regular interval
$(\alpha,\beta)$, then
$\hat{A}\coloneqq(\hat{\mathcal{A}},\Omega_{\alpha,\beta})$ is a regular
umbrae with regular interval includes $(-b,-a)$.

Fourier transform establishes the correspondence between regular umbrae.

Since we essentially only can deal with the regular umbrae, it is necessary to
establish a decomposition theorem for the singular one.

######  Theorem 2 (Component).

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a singular umbrae with index
$(\alpha,\beta)$. If there are regular umbrae pairs $(A_{1},A_{2})$ inducing
the decomposition $A=A_{1}[+]A_{2}$, then one of the regular umbrae $A_{i}$
has positive index $\alpha$, called the positive component of this
decomposition, and the other $A_{j}$ has negative index $\beta$, its opposite
$[-1]+A_{j}$ called the negative component of this decomposition.
$(A_{i},[-1]+A_{j})$ is called the regular decomposition of $A$.

######  Theorem 3 (Decomposition).

If $A=(\mathcal{A},\Omega_{a,b})$ is a umbrae with index $(\alpha,\beta)$,
then there are umbraes $A^{+},A^{-}$ such that

  1. 1.

$A^{+}=(\mathcal{A}^{+},\Omega_{a,b})$ is a regular umbrae with index
$(\alpha,\infty)$;

  2. 2.

$A^{-}=(\mathcal{A}^{-},\Omega_{a,b})$ is a regular umbrae with index
$(-\infty,\beta)$;

  3. 3.

$A=A^{+}[-]A^{-}$.

###### Proof.

We denoted

$\displaystyle\Lambda_{+}$
$\displaystyle=(\frac{1}{\mathrm{i}\sqrt{2\pi}}z^{-1}\mathrm{e}^{-z^{2}},\Omega_{0,\infty})=(\rho_{+},\Omega_{0,\infty});$
$\displaystyle\Lambda_{-}$
$\displaystyle=(\frac{1}{\mathrm{i}\sqrt{2\pi}}z^{-1}\mathrm{e}^{-z^{2}},\Omega_{-\infty,0})=(\rho_{-},\Omega_{-\infty,0}).$

According to Theorem 1,

  1. 1.

$\check{\Lambda}_{+}=(\check{\rho}_{+},\Omega_{-\infty,\infty})$ is a regular
umbrae with index $(0,\infty)$;

  2. 2.

$\check{\Lambda}_{-}=(\check{\rho}_{-},\Omega_{-\infty,\infty})$ is a regular
umbrae with index $(-\infty,0)$;

  3. 3.

$\check{\Lambda}_{+}[-]\check{\Lambda}_{-}=0$.

Finally, we can take
$A^{+}=A+\check{\Lambda}_{+},A^{-}=A+\check{\Lambda}_{-}$. ∎

This theorem allows us to transfer the problem of singular umbrae into regular
umbraes.

Next, we define a class of functions that the umbrae can be evaluated. This
function class will not be broad enough, but it is enough for our purposes.

######  Definition 2.6.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae, the index is
$(\alpha,\beta)$. $I$ is an open interval. If
$I\cap(\alpha,\beta)\neq\varnothing$, and the analytic function $f$ satisfies

  1. 1.

$f$ is an exponential type analytic function defined on
$\\{z\in\mathbb{C}:\mathrm{Re}z\in I\\}$;

  2. 2.

For every $t\in I\cap(\alpha,\beta)$, $\hat{\mathcal{A}}(z)f(\mathrm{i}z)$ is
Lebesgue integrable on $\mathbb{R}-\mathrm{i}t$,

then the class formed by such functions is denoted by $\mathcal{T}_{A}(I)$.

######  Definition 2.7 (Evaluation function).

If $f\in\mathcal{T}_{A}(I)$, then we defined

$f(A)\coloneqq\mathrm{eval}_{A}(f(z);z)\coloneqq\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\hat{\mathcal{A}}(z)f(\mathrm{i}z)\mathrm{d}z,$
(2.3)

where $t\in I\cap(\alpha,\beta)$.

Lemma 2.1 ensured the well-definedness of $f(A)$.

In fact, the argument from Lemma 2.1 can also tell us the following result
about summation method.

######  Proposition 2.1 (Gauss-Weierstrass summation method).

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae, the index is
$(\alpha,\beta)$. $f$ is an exponential type analytic function defined on
$\\{z\in\mathbb{C}:\mathrm{Re}z\in\tilde{I}\\}$. $I,\tilde{I}$ are open
intervals satisfied $I\subseteq\tilde{I}$.

If $f\in\mathcal{T}_{A}(I)$, then for every $t\in\tilde{I}\cap(\alpha,\beta)$
we have

$\lim\limits_{\varepsilon\to
0}\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\mathrm{e}^{-\varepsilon
z^{2}}\hat{\mathcal{A}}(z)f(\mathrm{i}z)\mathrm{d}z=f(A).$ (2.4)

###### Proof.

Because of

$\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\mathrm{e}^{-\varepsilon
z^{2}}\hat{\mathcal{A}}(z)f(\mathrm{i}z)\mathrm{d}z=\int_{-\infty-\mathrm{i}t_{0}}^{\infty-\mathrm{i}t_{0}}\mathrm{e}^{-\varepsilon
z^{2}}\hat{\mathcal{A}}(z)f(\mathrm{i}z)\mathrm{d}z,$

only need to let $\varepsilon\to 0$. ∎

We can also use (2.4) to define the evaluation function, but it will
complicate the limit calculus.

###  2.2 Well-definedness

Intuitively we have $f(A_{1}[-]A_{2})=f(A_{1})-f(A_{2})$, which inspired the
following proposition.

######  Proposition 2.2.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a singular umbrae with index
$(\alpha,\beta)$. $I$ is an open interval. If $I\supseteq[\beta,\alpha]$, and
the following conditions are satisfied,

  1. 1.

$(A_{1}^{+},A_{1}^{-}),(A_{2}^{+},A_{2}^{-})$ is the regular decomposition of
$A$;

  2. 2.

$f\in\left(\mathcal{T}_{A_{1}^{+}}(I)\cap\mathcal{T}_{A_{1}^{-}}(I)\right)\cap\left(\mathcal{T}_{A_{2}^{+}}(I)\cap\mathcal{T}_{A_{2}^{-}}(I)\right)$,

then

  1. 1.

$f(A_{1}^{+})-f(A_{1}^{-})=f(A_{2}^{+})-f(A_{2}^{-})$;

  2. 2.

$A_{0}=A_{1}^{+}[-]A_{2}^{+}=A_{1}^{-}[-]A_{2}^{-}$ is a regular umbrae;

  3. 3.

$f\in\mathcal{T}_{A_{0}}(I)$.

###### Proof.

According to Theorem 2, the regular interval of the component of the regular
decomposition always intersects with $I$, and
$A_{0}=A_{1}^{+}[-]A_{2}^{+}=A_{1}^{-}[-]A_{2}^{-}$ is a regular umbrae.

Notice that

$\displaystyle f\in\mathcal{T}_{A_{1}^{+}}(I)\cap\mathcal{T}_{A_{2}^{+}}(I)$
$\displaystyle\Rightarrow
f\in\mathcal{T}_{A_{0}}(I\cap(\alpha,\alpha+\varepsilon))$ $\displaystyle
f\in\mathcal{T}_{A_{1}^{-}}(I)\cap\mathcal{T}_{A_{2}^{-}}(I)$
$\displaystyle\Rightarrow
f\in\mathcal{T}_{A_{0}}(I\cap(\beta-\varepsilon,\beta)).$

Since $f$ is exponential type,
$f(A_{1}^{+})-f(A_{2}^{+})=f(A_{1}^{-})-f(A_{2}^{-})$ holds from Lemma 2.1.
Lemma 2.2 ensured that $f\in\mathcal{T}_{A_{0}}(I)$. ∎

######  Definition 2.8.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a singular umbrae with index
$(\alpha,\beta)$. $I$ is an open interval includes $[\beta,\alpha]$.

If there is a regular decomposition $(A^{+},A^{-})$ of $A$ such that
$f\in\mathcal{T}_{A^{+}}(I)\cap\mathcal{T}_{A^{-}}(I)$, then we defined
$f(A)\coloneqq f(A^{+})-f(A^{-})$.

Here we are not going to give a complete definition of $f(A)$ for the singular
umbrae $A$. The general form of Proposition 2.2 can be derived by Lemma 2.1
and Theorem 3.

######  Proposition 2.3.

Suppose
$A_{1}=(\mathcal{A}_{1},\Omega_{a_{1},b_{1}}),A_{2}=(\mathcal{A}_{2},\Omega_{a_{2},b_{2}})$
are regular umbrae with index $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})$.

If $f(z_{1},z_{2})$ satisfied

  1. 1.

For every $\mathrm{Re}z_{2}\in I_{2}$ we have
$f(\cdot,z_{2})\in\mathcal{T}_{A_{1}}(I_{1})$;

  2. 2.

$f(A_{1},\cdot)\in\mathcal{T}_{A_{2}}(I_{2})$;

  3. 3.

For every $\mathrm{Re}z_{1}\in I_{1}$ we have
$f(z_{1},\cdot)\in\mathcal{T}_{A_{2}}(I_{2})$;

  4. 4.

$f(\cdot,A_{2})\in\mathcal{T}_{A_{1}}(I_{1})$;

  5. 5.

There exists $t_{1}\in I_{1}\cap(\alpha_{1},\beta_{1}),t_{2}\in
I_{2}\cap(\alpha_{2},\beta_{2})$ such that
$\hat{\mathcal{A}}_{1}(z_{1})\hat{\mathcal{A}}_{2}(z_{2})f(\mathrm{i}z_{1},\mathrm{i}z_{2})$
is Lebesgue integrable on
$(\mathbb{R}-\mathrm{i}t_{1})\times(\mathbb{R}-\mathrm{i}t_{2})$,

then we have

$\mathrm{eval}_{A_{2}}(\mathrm{eval}_{A_{1}}(f(z_{1},z_{2});z_{1});z_{2})=\mathrm{eval}_{A_{1}}(\mathrm{eval}_{A_{2}}(f(z_{1},z_{2});z_{2});z_{1}),$
(2.5)

i.e. $f(A_{1},A_{2})$ is well-defined.

###### Proof.

Apply the Fubini theorem. ∎

######  Proposition 2.4.

Suppose
$A_{1}=(\mathcal{A}_{1},\Omega_{a,b}),A_{2}=(\mathcal{A}_{2},\Omega_{a,b})$
are regular umbrae with index $(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{2})$.

If $g(z_{1},z_{2})=f(z_{1}+z_{2})$ satisfies the conditions of Proposition
2.3, and there is an open interval $I\subseteq I_{1}+I_{2}$ such that
$f\in\mathcal{T}_{A_{1}+A_{2}}(I)$, then

$f(A_{1}+A_{2})=g(A_{1},A_{2}),$

i.e. $f(A_{1}+A_{2})$ is well-defined.

###### Proof.

We assume that $s\in(a,b)$. First notice that

$\displaystyle g(A_{1},A_{2})$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty-\mathrm{i}t_{1}}^{\infty-\mathrm{i}t_{1}}\int_{-\infty-\mathrm{i}t_{2}}^{\infty-\mathrm{i}t_{2}}\hat{\mathcal{A}}_{1}(z_{1})\hat{\mathcal{A}}_{2}(z_{2})f(\mathrm{i}z_{1}+\mathrm{i}z_{2})\mathrm{d}z_{2}\mathrm{d}z_{1}$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty-\mathrm{i}(t_{1}+t_{2})}^{\infty-\mathrm{i}(t_{1}+t_{2})}\int_{-\infty-\mathrm{i}t_{2}}^{\infty-\mathrm{i}t_{2}}\mathrm{e}^{s(z_{0}-z_{2})}\hat{\mathcal{A}}_{1}(z_{0}-z_{2})\mathrm{e}^{sz_{2}}\hat{\mathcal{A}}_{2}(z_{2})\mathrm{e}^{-sz_{0}}f(\mathrm{i}z_{0})\mathrm{d}z_{2}\mathrm{d}z_{0}$
$\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty-\mathrm{i}(t_{1}+t_{2})}^{\infty-\mathrm{i}(t_{1}+t_{2})}\mathrm{e}^{-sz_{0}}(\mathrm{e}^{s(\cdot)}\hat{\mathcal{A}}_{1}\ast\mathrm{e}^{s(\cdot)}\hat{\mathcal{A}}_{2})(z_{0})f(\mathrm{i}z_{0})\mathrm{d}z_{0},$

and then notice that $f$ is an exponential type analytic function defined on
$\\{z\in\mathbb{C}:\mathrm{Re}z\in I_{1}+I_{2}\\}$, so the conclusion followed
by Lemma 2.1. ∎

######  Proposition 2.5.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae with index
$(\alpha,\beta)$, $r\in\mathbb{R}$.

If $g(z)=f(rz)$, then we have

$f\in\mathcal{T}_{rA}(I)\Leftrightarrow g\in\mathcal{T}_{A}(r^{-1}I),$

meanwhile $g(A)=f(rA)$ i.e. $f(rA)$ is well-defined.

Finally, we will be concerned about the well-definedness related to some
special singular umbrae, such as $(c)$ and $\mathrm{D}$. We hope they act like
a constant and derivative.

######  Proposition 2.6.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae with index
$(\alpha,\beta)$, $c\in\mathbb{C}$.

If $g(z)=f(z+c)$, then we have

$f\in\mathcal{T}_{A+(c)}(I)\Leftrightarrow g\in\mathcal{T}_{A}(I-c),$

meanwhile $g(A)=f(A+(c))$ i.e. $f(A+c)$ is well-defined.

Unfortunately, the derivative calculus is not completely well-defined.
However, understanding this counterexample can also increase our understanding
of umbrae calculus. Consider
$f(z)=1,A=(z^{-1}\mathrm{e}^{-z^{2}},\Omega_{0,\infty})$, we have
$f\in\mathcal{T}_{A+\mathrm{D}}(\mathbb{R}),f^{\prime}\in\mathcal{T}_{A}(\mathbb{R})$,
but $f(A+\mathrm{D})=1,f^{\prime}(A)=0$.

######  Theorem 4.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae with index
$(\alpha,\beta)$.

If
$\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}|\hat{\mathcal{A}}(z)f(\mathrm{i}z+z_{0})|\mathrm{d}z$
is uniformly bounded for sufficiently small $z_{0}$, and
$f(\cdot+z_{0})\in\mathcal{T}_{A}(I),t\in I$, then we have

  1. 1.

$f^{\prime}\in\mathcal{T}_{A}(I)\Rightarrow
f^{\prime}(A)=\frac{\mathrm{d}}{\mathrm{d}z_{0}}f(A+z_{0})(0)$;

  2. 2.

$f\in\mathcal{T}_{A+\mathrm{D}}(I)\Rightarrow
f(A+\mathrm{D})=\frac{\mathrm{d}}{\mathrm{d}z_{0}}f(A+z_{0})(0)$,

i.e. $f(A+\mathrm{D})$ is well-defined.

###### Proof.

Suppose that $f^{\prime}\in\mathcal{T}_{A}(I)$. For

$g_{\varepsilon}(z_{0})=\frac{1}{\sqrt{2\pi}}\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\mathrm{e}^{-\varepsilon
z^{2}}\hat{\mathcal{A}}(z)f(\mathrm{i}z+z_{0})\mathrm{d}z,$

we have

  1. 1.

$\lim\limits_{\varepsilon\to 0}g_{\varepsilon}(z_{0})=f(A+z_{0})$;

  2. 2.

Sequence $(g_{\varepsilon})_{\varepsilon>0}$ is uniformly bounded.

By the Montel’s theorem, there exists a subsequence
$(g_{\varepsilon_{n}})_{n\in\mathbb{N}}$ locally uniformly converges to
$f(A+z_{0})$. $f^{\prime}$ is again an exponential type analytic function, by
the Leibniz integral rule we have

$g^{\prime}_{\varepsilon}(z_{0})=\frac{1}{\sqrt{2\pi}}\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\mathrm{e}^{-\varepsilon
z^{2}}\hat{\mathcal{A}}(z)f^{\prime}(\mathrm{i}z+z_{0})\mathrm{d}z.$

Because $g^{\prime}_{\varepsilon}(0)\to
f^{\prime}(A),g^{\prime}_{\varepsilon_{n}}(0)\to\frac{\mathrm{d}}{\mathrm{d}z_{0}}f(A+z_{0})(0).$

Suppose that $f\in\mathcal{T}_{A+\mathrm{D}}(I)$, and $t+z_{0}\in I$. Only
need to notice that

$\displaystyle\sqrt{2\pi}\tilde{g}_{\varepsilon}(z_{0})$
$\displaystyle=\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\mathrm{e}^{-\varepsilon(z-\mathrm{i}z_{0})^{2}}\hat{\mathcal{A}}(z)f(\mathrm{i}z+z_{0})\mathrm{d}z$
$\displaystyle=\int_{-\infty-\mathrm{i}(t+z_{0})}^{\infty-\mathrm{i}(t+z_{0})}\mathrm{e}^{-\varepsilon
z^{2}}\hat{\mathcal{A}}(z+\mathrm{i}z_{0})f(\mathrm{i}z)\mathrm{d}z$
$\displaystyle=\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\mathrm{e}^{-\varepsilon
z^{2}}\hat{\mathcal{A}}(z+\mathrm{i}z_{0})f(\mathrm{i}z)\mathrm{d}z,$

and we still have

  1. 1.

$\lim\limits_{\varepsilon\to 0}\tilde{g}_{\varepsilon}(z_{0})=f(A+z_{0})$;

  2. 2.

Sequence $(\tilde{g}_{\varepsilon})_{\varepsilon>0}$ is uniformly bounded.

By the Leibniz integral rule again, we have

$\tilde{g}^{\prime}_{\varepsilon}(z_{0})=\frac{1}{\sqrt{2\pi}}\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}\mathrm{e}^{-\varepsilon
z^{2}}\mathrm{i}\hat{\mathcal{A}}^{\prime}(z+\mathrm{i}z_{0})f(\mathrm{i}z)\mathrm{d}z,$

this completes the proof. ∎

This method seems to be able to replace $\mathrm{D}$ with an umbrae whose
dominating interval is also $\mathbb{R}$. In fact, Theorem 5 will strengthen
the condition to prove more general cases.

###  2.3 Specific calculation

After establishing the required well-definedness, we can see how various
conclusions emerge naturally.

######  Theorem 5.

Suppose $(f(\mathrm{i}z),\Omega_{\beta,\alpha})$ is a umbrae with index
$(b,a)$.

If $A_{k}=(\mathcal{A}_{k},\Omega_{a_{k},b_{k}})$ is a umbrae with index
$(\alpha_{k},\beta_{k})$, satisfied

  1. 1.

$a_{k}<a$; $b<b_{k}$;

  2. 2.

$\beta_{k}-\alpha_{k}>-(\alpha-\beta)$,

then we have

  1. 1.

$(f(A_{k}+\mathrm{i}z),\Omega_{\beta-\beta_{k},\alpha-\alpha_{k}})$ is a
umbrae with index at least $(b\vee a_{k},a\wedge b_{k})$;

  2. 2.

If $(a_{k},b_{k})\cap(a_{j},b_{j})\neq\varnothing$, and
$(\beta_{k}-\alpha_{k})+(\beta_{j}-\alpha_{j})>-(\alpha-\beta)$ then
$f(A_{k}+A_{j}+\mathrm{i}z)$ is well-defined in the sense of Definition 2.4;

  3. 3.

If $(a_{k},b_{k})\cap(a_{j},b_{j})\neq\varnothing$, and
$(\beta-\beta_{k},\alpha-\alpha_{k})\cap(\beta-\beta_{j},\alpha-\alpha_{j})\neq\varnothing$
then $f((A_{k}[+]A_{j})+\mathrm{i}z)$ is well-defined, satisfied

$f((A_{k}[+]A_{j})+\mathrm{i}z)=f(A_{k}+\mathrm{i}z)+f(A_{j}+\mathrm{i}z).$
(2.6)

###### Proof.

First assume that $\alpha_{k}<\beta_{k}$, and take
$a_{k}<\tilde{a}_{k}<\tilde{a}<a,b<\tilde{b}<\tilde{b}_{k}<b_{k}$, we have

$\displaystyle|\hat{\mathcal{A}}(z)|$
$\displaystyle\leq\begin{cases*}C_{1}(-\mathrm{Im}z)\cdot\mathrm{e}^{-\tilde{b}_{k}\mathrm{Re}z}&$\mathrm{Re}z\geq
0,\alpha_{k}<-\mathrm{Im}z<\beta_{k}$\\\
C_{1}(-\mathrm{Im}z)\cdot\mathrm{e}^{-\tilde{a}_{k}\mathrm{Re}z}&$\mathrm{Re}z\leq
0,\alpha_{k}<-\mathrm{Im}z<\beta_{k}$\end{cases*}$
$\displaystyle|f(\mathrm{i}z)|$
$\displaystyle\leq\begin{cases*}C_{2}(-\mathrm{Im}z)\cdot\mathrm{e}^{\tilde{b}\mathrm{Re}z}&$\mathrm{Re}z\geq
0,\beta<-\mathrm{Im}z<\alpha$\\\
C_{2}(-\mathrm{Im}z)\cdot\mathrm{e}^{\tilde{a}\mathrm{Re}z}&$\mathrm{Re}z\leq
0,\beta<-\mathrm{Im}z<\alpha$\end{cases*}$

So we can directly verify that, for
$\mathrm{Re}z_{0}\in(\beta-\beta_{k},\alpha-\alpha_{k})$ there exists $t$ such
that

$\displaystyle|f(A+z_{0})|$
$\displaystyle\leq\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}|\hat{\mathcal{A}}(z)|\cdot|f(\mathrm{i}z+z_{0})|\mathrm{d}z$
$\displaystyle\lesssim\mathrm{e}^{\tilde{b}\mathrm{Im}z_{0}}\int_{0}^{\infty}\mathrm{e}^{-(\tilde{b}_{k}-\tilde{b})x}\mathrm{d}x+\mathrm{e}^{\tilde{b}\mathrm{Im}z_{0}}\int_{-\mathrm{Im}z_{0}}^{0}\mathrm{e}^{(\tilde{b}-\tilde{a}_{k})x}\mathrm{d}x+$
$\displaystyle\phantom{==}\mathrm{e}^{\tilde{a}\mathrm{Im}z_{0}}\int_{-\infty}^{-\mathrm{Im}z_{0}}\mathrm{e}^{(\tilde{a}-\tilde{a}_{k})x}\mathrm{d}x$
$\displaystyle\leq\frac{\mathrm{e}^{\tilde{b}\mathrm{Im}z_{0}}}{\tilde{b}_{k}-\tilde{b}}+\frac{\mathrm{e}^{\tilde{b}\mathrm{Im}z_{0}}-\mathrm{e}^{\tilde{a}_{k}\mathrm{Im}z_{0}}}{\tilde{b}-\tilde{a}_{k}}+\frac{\mathrm{e}^{\tilde{a}_{k}\mathrm{Im}z_{0}}}{\tilde{a}-\tilde{a}_{k}}\quad\mathrm{Im}z_{0}\geq
0.$

Repeat the same method to estimate the other half and we can conclude that
$(f(A+\mathrm{i}z),\Omega_{\beta-\beta_{k},\alpha-\alpha_{k}})$ is a umbrae
with index $(b\vee a_{k},a\wedge b_{k})$. Finally, apply Theorem 3 and
Definition 2.8 to $A$. ∎

The condition of the umbrae under Theorem 5 seems to be the most natural, but
not enough for our purposes. We can also notice that the performance of the
umbrae is more determined by its dominating interval.

In order to be able to perform basic calculations, we need the following
theorem.

######  Lemma 2.3.

If $(f(\mathrm{i}z),\Omega_{\beta,\alpha})$ is a regular umbrae with index
$(b,a)$ satisfied $0\in(b,a)$, then for every $s\in(\beta,\alpha)$ we have
$f(s+\mathrm{i}\xi)\in\mathcal{S}(\mathbb{R})$.

###### Proof.

Apply Theorem 1. ∎

######  Lemma 2.4.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae with index
$(\alpha,\beta)$, and

  1. 1.

$-\infty<\alpha<\beta$;

  2. 2.

For every $t\in(a,b)$ we have $\mathrm{e}^{-\alpha
x}\mathcal{A}(x-\mathrm{i}t)\in\mathcal{S}^{\prime}(\mathbb{R})$;

  3. 3.

Denote
$h(\xi)\coloneqq\mathrm{e}^{-t\xi}\mathcal{F}(\mathrm{e}^{-\alpha(x-\mathrm{i}t)}\mathcal{A}(x-\mathrm{i}t))$
in the sense of $\mathcal{S}^{\prime}(\mathbb{R})$.

If $(f(\mathrm{i}z),\Omega_{\beta_{0},\alpha_{0}})$ is a umbrae with index
$(b_{0},a_{0})$ satisfied

$a<a_{0},\quad b_{0}<b,$

then for every $s\in(\beta_{0},\alpha_{0})$ we have

$\displaystyle f(A+\alpha-s)$
$\displaystyle=\int_{\mathbb{R}}\mathrm{e}^{t\xi}h(\xi)\mathrm{e}^{-t\xi}f(s+\mathrm{i}\xi)\mathrm{d}\xi.$

###### Proof.

According to the continuity of Fourier transform on
$\mathcal{S}^{\prime}(\mathbb{R})$, for every $t\in(a,b)$ we have

$\lim\limits_{\varepsilon\to
0^{+}}\mathrm{e}^{t\xi}\hat{\mathcal{A}}(\xi-\mathrm{i}(\alpha+\varepsilon))=\mathrm{e}^{t\xi}h(\xi),$
(2.7)

in the sense of $\mathcal{S}^{\prime}(\mathbb{R})$.

Apply Theorem 3 on $f$ to get

  1. 1.

$(f^{+}(\mathrm{i}z),\Omega_{\beta_{0},\alpha_{0}})$ is a regular umbrae with
index $(b_{0},\infty)$;

  2. 2.

$(f^{-}(\mathrm{i}z),\Omega_{\beta_{0},\alpha_{0}})$ is a regular umbrae with
index $(-\infty,a_{0})$;

  3. 3.

$f=f^{+}-f^{-}$,

according to Lemma 2.3, for every $s\in(\beta_{0},\alpha_{0})$ we have

$\displaystyle\int_{\mathbb{R}}\hat{\mathcal{A}}(\xi-\mathrm{i}(\alpha+\varepsilon))f^{+}(s+\mathrm{i}\xi)\mathrm{d}\xi$
$\displaystyle=\int_{\mathbb{R}}\mathrm{e}^{\tilde{b}\xi}\hat{\mathcal{A}}(\xi-\mathrm{i}(\alpha+\varepsilon))\mathrm{e}^{-\tilde{b}\xi}f^{+}(s+\mathrm{i}\xi)\mathrm{d}\xi$
$\displaystyle\to\int_{\mathbb{R}}\mathrm{e}^{\tilde{b}\xi}h(\xi)\mathrm{e}^{-\tilde{b}\xi}f^{+}(s+\mathrm{i}\xi)\mathrm{d}\xi$
$\displaystyle=\int_{\mathbb{R}}\mathrm{e}^{t\xi}h(\xi)\mathrm{e}^{-t\xi}f^{+}(s+\mathrm{i}\xi)\mathrm{d}\xi,$

Repeat the same method to the other half and we can conclude that

$\displaystyle\int_{\mathbb{R}}\hat{\mathcal{A}}(\xi-\mathrm{i}(\alpha+\varepsilon))f(s+\mathrm{i}\xi)\mathrm{d}\xi$
$\displaystyle\to\int_{\mathbb{R}}\mathrm{e}^{t\xi}h(\xi)\mathrm{e}^{-t\xi}f(s+\mathrm{i}\xi)\mathrm{d}\xi$
$\displaystyle f(A+(\alpha+\varepsilon)-s)$ $\displaystyle\to f(A+\alpha-s),$

where $a<\tilde{a}<a_{0},b_{0}<\tilde{b}<b,a<t<b$. The continuity of $f(A+z)$
comes from Theorem 5. ∎

######  Theorem 6.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a umbrae with index
$(\alpha,\beta)$, and

  1. 1.

$-\infty<\alpha\leq\beta$;

  2. 2.

For every $t\in(a,b)$ we have $\mathrm{e}^{-\alpha
x}\mathcal{A}(x-\mathrm{i}t)\in\mathcal{S}^{\prime}(\mathbb{R})$;

  3. 3.

Denote
$h(\xi)\coloneqq\mathrm{e}^{-t\xi}\mathcal{F}(\mathrm{e}^{-\alpha(x-\mathrm{i}t)}\mathcal{A}(x-\mathrm{i}t))$
in the sense of $\mathcal{S}^{\prime}(\mathbb{R})$,

If $(f(\mathrm{i}z),\Omega_{\beta_{0},\alpha_{0}})$ is a umbrae with index
$(b_{0},a_{0})$ satisfied

  1. 1.

$a<a_{0}$; $b_{0}<b$;

  2. 2.

$\alpha\in(\beta_{0},\alpha_{0})$,

then we have

$\displaystyle f(A)$
$\displaystyle=\int_{\mathbb{R}}\mathrm{e}^{t\xi}h(\xi)\mathrm{e}^{-t\xi}f(\alpha+\mathrm{i}\xi)\mathrm{d}\xi.$
(2.8)

###### Proof.

For the singular umbrae $A$, only need to apply Theorem 3. ∎

######  Corollary 2.1.

Suppose $f$ is an exponential type analytic function defined on
$\\{z\in\mathbb{C}:\mathrm{Re}z\in(\beta,\alpha)\\}$.

  1. 1.

$f((c)+z)=f(c+z)$;

  2. 2.

$f(\mathrm{D}+z)=f^{\prime}(z)$;

  3. 3.

$f(\Delta+z)=f(z+1)-f(z)$, where $\beta-\alpha>1$.

######  Proposition 2.7.

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a umbrae with index
$(\alpha,\beta)$.

  1. 1.

$\mathrm{e}^{Az}=\mathcal{A}(z)\quad z\in\Omega_{a,b}$;

  2. 2.

$A^{n}=\mathcal{A}^{(n)}(0)\quad 0\in(a,b),n\in\mathbb{N}$;

  3. 3.

$(A^{\mathrm{i}z},\Omega_{-\infty,\infty})$ is a regular umbrae with index
$(-\pi,\pi)$, where $0\in(a,b),\beta>0$.

###### Proof.

For the singular umbrae $A$, only need to apply Theorem 3. ∎

Just like Carlson’s theorem shows, the function class formed by such $A^{z}$
is also determined by the value on natural numbers.

###  2.4 Limit calculus

Next, we need several important lemmas to help us calculate $f(A)$
specifically.

######  Lemma 2.5.

Suppose $f$ is locally integrable, $b>0,n\geq 0$.

If $\mathrm{e}^{-bx}(1+|x|)^{n}f(x)\in L^{1}(0,\infty)$, then
$\mathrm{e}^{-bx}(1+|x|)^{n}\int_{0}^{x}f(t)\mathrm{d}t\in L^{1}(0,\infty)$.

###### Proof.

Only need to notice that

$\displaystyle\mathrm{e}^{-bx}(1+x)^{n}\int_{0}^{x}|f(t)|\mathrm{d}t$
$\displaystyle\leq\int_{0}^{x}\mathrm{e}^{-bt}(1+t)^{n}|f(t)|(1+x-t)^{n}\mathrm{e}^{-b(x-t)}\mathrm{d}t$
$\displaystyle=(\mathrm{e}^{-b(\cdot)}(1+\cdot)^{n}f(\cdot)\ast\mathrm{e}^{-b(\cdot)}(1+\cdot)^{n})(x).$

The result is obtained from Young’s inequality. ∎

The umbrae that satisfy the following properties are good enough to give a
more accurate estimate for $\hat{\mathcal{A}}$, and Bernoulli umbrae
$\mathrm{B}$ is such a umbrae. Ultimately, it will allow us to do calculus
$f(A)$ on as many functions as possible. In fact, if there is no such fine
result, we will not be able to process the Gosper series as we promised.

######  Lemma 2.6 (Good umbrae).

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a regular umbrae with index
$(\alpha,\beta)$.

  1. 1.

If $b<\infty$ and
$(\mathcal{A}(z)\prod_{j=1}^{m}(z-b_{j})^{l_{j}},\Omega_{a,b+\varepsilon})$ is
a umbrae, then we have

$\displaystyle|\hat{\mathcal{A}}(\xi-\mathrm{i}t)|\leq
C(t)\mathrm{e}^{-b\xi}(1+|\xi|)^{l-1}\quad\xi\geq 0,$ (2.9)

where $l=\max\\{l_{1},\ldots,l_{m}\\},-\mathrm{Im}b_{j}=b$;

  2. 2.

If $a>-\infty$ and
$(\mathcal{A}(z)\prod_{j=1}^{n}(z-a_{j})^{k_{j}},\Omega_{a-\varepsilon,b})$ is
a umbrae, then we have

$\displaystyle|\hat{\mathcal{A}}(\xi-\mathrm{i}t)|\leq
C(t)\mathrm{e}^{-a\xi}(1+|\xi|)^{k-1}\quad\xi\leq 0,$ (2.10)

where $k=\max\\{k_{1},\ldots,k_{n}\\},-\mathrm{Im}a_{j}=a$.

###### Proof.

Consider the Fourier transform of the singular part of $\hat{\mathcal{A}}$ at
the pole ∎

It seems that this lemma can be generalized further, but the current form is
sufficient for us.

######  Theorem 7 (Limit calculus).

Suppose $A$ is a regular umbrae, $(f_{n})_{n\in\mathbb{N}}$ is a sequence in
$\mathcal{T}_{A}(I)$, and $f\in\mathcal{T}_{A}(I),t\in I$.

If $\hat{\mathcal{A}}(z)f_{n}(\mathrm{i}z)$ tends to
$\hat{\mathcal{A}}(z)f(\mathrm{i}z)$ in the sense of
$L^{1}(\mathbb{R}-\mathrm{i}t)$, then

$\lim\limits_{n\to\infty}f_{n}(A)=f(A).$ (2.11)

######  Corollary 2.2 (Interchangeability).

Suppose $A=(\mathcal{A},\Omega_{a,b})$ is a umbrae with index
$(\alpha,\beta)$, and satisfy $0\in(a,b)$.

For $f(z)=\sum_{n=0}^{\infty}\frac{a_{n}z^{n}}{n!}$ we have

  1. 1.

If there exists $r<(-a)\vee b$ such that

$\sum_{n=0}^{\infty}|a_{n}|r^{-n}<\infty,$ (2.12)

then $f(A)=\sum_{n=0}^{\infty}\frac{a_{n}A^{n}}{n!}$;

  2. 2.

If $A$ satisfied one of the conditions in Lemma 2.6, and $f$ satisfies

$\displaystyle\sum_{n=0}^{\infty}|b|^{-n}n^{l-1}|a_{n}|$
$\displaystyle<\infty$
$\displaystyle\sum_{n=0}^{\infty}|a|^{-n}n^{k-1}|a_{n}|$
$\displaystyle<\infty,$

then $f(A)=\sum_{n=0}^{\infty}\frac{a_{n}A^{n}}{n!}$.

###### Proof.

For the singular umbrae $A$, only need to apply Theorem 3. ∎

This corollary will eventually provide us the interchangeable results which we
care about.

######  Definition 2.9 (Hierarchy).

If $f$ is defined on $\\{z\in\mathbb{C}:\mathrm{Re}z>0\\}$ and satisfies the
following properties

  1. 1.

For $\mathrm{Re}t>0$ we have

$\displaystyle|f(t+\mathrm{i}\xi)|\mathrm{e}^{-b\xi}(1+|\xi|)^{l}$
$\displaystyle\in L^{1}((0,\infty);\xi)$
$\displaystyle\lim\limits_{\mathrm{Re}t\to\infty}\int_{-\infty}^{\infty}\left|f(t+\mathrm{i}\xi)-\sum_{k=0}^{p-1}\frac{f^{(k)}(t)}{k!}(\mathrm{i}\xi)^{k}\right|\mathrm{e}^{-b\xi}(1+|\xi|)^{l}\mathrm{d}\xi$
$\displaystyle=0,$

locally uniformly with respect to $\mathrm{Im}t$;

  2. 2.

For $\mathrm{Re}t>0$ we have

$\displaystyle|f(t+\mathrm{i}\xi)|\mathrm{e}^{-a\xi}(1+|\xi|)^{k}$
$\displaystyle\in L^{1}((-\infty,0);\xi)$
$\displaystyle\lim\limits_{\mathrm{Re}t\to\infty}\int_{-\infty}^{\infty}\left|f(t+\mathrm{i}\xi)-\sum_{k=0}^{p-1}\frac{f^{(k)}(t)}{k!}(\mathrm{i}\xi)^{k}\right|\mathrm{e}^{-a\xi}(1+|\xi|)^{k}\mathrm{d}\xi$
$\displaystyle=0,$

locally uniformly with respect to $\mathrm{Im}t$;

  3. 3.

$|f(z)|\leq C(\mathrm{Re}z)\cdot\mathrm{e}^{s|\mathrm{Im}z|}$, where $C$ is
locally bounded,

then the space formed by such functions is denoted by
$\mathcal{T}_{a,k;b,l}^{(p-1)}$.

In fact, we only need $t$ to go to infinity along the natural number.

######  Theorem 8 (Hierarchy).

If $0\in(a,b)$ and $k,l\geq 0$, then

$f(z)\in\mathcal{T}_{a,k;b,l}^{(p-1)}\Rightarrow\int_{1}^{x}f(t)\mathrm{d}t\in\mathcal{T}_{a,k;b,l}^{(p)}.$
(2.13)

###### Proof.

Denote $F(z)=\int_{1}^{x}f(t)\mathrm{d}t$. Notice that

$\displaystyle\left\lVert\mathrm{e}^{-b\xi}(1+|\xi|)^{l}\left(F(t+\mathrm{i}\xi)-\sum_{k=0}^{p}\frac{F^{(k)}(t)}{k!}(\mathrm{i}\xi)^{k}\right)\right\rVert_{L^{1}(0,\infty)}$
$\displaystyle=$
$\displaystyle\left\lVert\mathrm{e}^{-b\xi}(1+|\xi|)^{l}\int_{0}^{\xi}\left(f(t+\mathrm{i}\xi_{0})-\sum_{k=0}^{p-1}\frac{f^{(k)}(t)}{k!}(\mathrm{i}\xi_{0})^{k}\right)\mathrm{d}\xi_{0}\right\rVert_{L^{1}(0,\infty)}$
$\displaystyle\leq$
$\displaystyle\left\lVert\mathrm{e}^{-b\xi}(1+|\xi|)^{l}\right\rVert_{L^{1}(0,\infty)}\cdot\left\lVert\mathrm{e}^{-b\xi}(1+|\xi|)^{l}\left(f(t+\mathrm{i}\xi)-\sum_{k=0}^{p-1}\frac{f^{(k)}(t)}{k!}(\mathrm{i}\xi)^{k}\right)\right\rVert_{L^{1}(0,\infty)}$
$\displaystyle\to 0,$

Repeat the same method to estimate the other half and we can conclude that (8)
holds. ∎

Similar to Theorem 6, this theorem will help us calculate $f(\mathrm{B})$ in a
more direct way.

##  3 Application

Let’s first redeem the promise and see how the MS-fractional summation related
to the Bernoulli umbrae.

Although the selection of $\Omega_{a,b}$ does not need to include the origin,
here we still only care about the Bernoulli umbrae that includes the origin,
which allows us to define $\mathrm{B}^{n}$. Some conclusions can be
generalized to other Bernoulli umbras.

######  Lemma 3.1.

$\mathrm{B}+\Delta=1+\mathrm{D}.$

######  Theorem 9.

If there exists $p\in\mathbb{N}$ such that

  1. 1.

$f\in\mathcal{T}_{-2\pi,0;2\pi,0}^{(p-1)}$;

  2. 2.

$\lim\limits_{t\to\infty}f^{(p)}(t+\mathrm{i}\xi)=0$ locally uniformly,

then for every $\mathrm{Re}z>-1$ we have

$(\mathrm{MS})\text{-}\sum_{k=1}^{z}f(k)=\int_{\mathrm{B}}^{\mathrm{B}+z}f(k)\mathrm{d}k.$
(3.1)

###### Proof.

Denote $F(z)=\int_{1}^{x}f(t)\mathrm{d}t$. Notice that the Bernoulli umbrae
$\mathrm{B}$ satisfies the condition of Lemma 2.6, therefore according to
Definition 2.7 and 2.9, $F(t+\mathrm{B})$ is defined for $\mathrm{Re}t>0$.

By Theorem 8, we have

$\displaystyle\lim\limits_{n\to\infty}\left(F(n+\mathrm{B})-\sum_{k=0}^{p}\frac{f^{(k-1)}(n)\mathrm{B}_{k}}{k!}\right)=0.$

By the Lagrangian remainder theorem, we have

$\sum_{k=0}^{p}\frac{(f^{(k-1)}(t+y)-f^{(k-1)}(t))\mathrm{B}_{k}}{k!}=\sum_{k=0}^{p}a_{k}(t)y^{k}+o(1)y^{p+1},$

therefore we have polynomial $P_{n}(y)=\sum_{k=0}^{p}a_{k}(n)y^{k}$ such that

$\displaystyle\lim\limits_{n\to\infty}\left(\sum_{k=1}^{n}(f(k+y)-f(k))-P_{n}(y)\right)$
$\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}((F(n+y+\mathrm{B})-F(n+\mathrm{B}))-P_{n}(y))-\int_{\mathrm{B}}^{\mathrm{B}+y}f(k)\mathrm{d}k$
$\displaystyle=$
$\displaystyle\lim\limits_{n\to\infty}o(1)y^{p+1}-\int_{\mathrm{B}}^{\mathrm{B}+y}f(k)\mathrm{d}k$
$\displaystyle=-\int_{\mathrm{B}}^{\mathrm{B}+y}f(k)\mathrm{d}k.$

Finally, by Definition 2.8 and Corollary 2.1, we have

$\displaystyle\mathrm{B}+\Delta=1+\mathrm{D}$
$\displaystyle\Rightarrow(\mathrm{B}+1)[-]\mathrm{B}=1+\mathrm{D}$
$\displaystyle\Rightarrow
F(y+((\mathrm{B}+1)[-]\mathrm{B}))=F(y+1+\mathrm{D})$
$\displaystyle\Rightarrow F(y+\mathrm{B}+1)-F(t+\mathrm{B})=F^{\prime}(y+1),$

which satisfies the conditions in Definition 1.1. ∎

######  Corollary 3.1.

If

$f(z)=\sum_{m=1}^{M}b_{m}z^{-m}+\sum_{n=0}^{\infty}\frac{a_{n}}{n!}z^{n}$

and there exists $p\in\mathbb{N}$ satisfies that

  1. 1.

$f\in\mathcal{T}_{-2\pi,0;2\pi,0}^{(p-1)}$;

  2. 2.

$\lim\limits_{t\to\infty}f^{(p)}(t+\mathrm{i}\xi)=0$ locally uniformly;

  3. 3.

$\sum_{n=0}^{\infty}(2\pi)^{-n}|a_{n}|<\infty$,

then for every $\mathrm{Re}\alpha,\mathrm{Re}\beta>-1$ we have

$(\mathrm{MS})\text{-}\sum_{k=\alpha+1}^{\beta}f(k)=\sum_{m=1}^{M}\left((\mathrm{MS})\text{-}\sum_{k=\alpha+1}^{\beta}b_{m}z^{-m}\right)+\sum_{n=0}^{\infty}\left((\mathrm{MS})\text{-}\sum_{k=\alpha+1}^{\beta}\frac{a_{n}}{n!}z^{n}\right).$
(3.2)

###### Proof.

Apply Corollary 2.2 and Theorem 9. ∎

######  Corollary 3.2.

If there exists $p\in\mathbb{N}$ satisfies that

  1. 1.

$f^{\prime}\in\mathcal{T}_{-2\pi,0;2\pi,0}^{(p-1)}$;

  2. 2.

$\lim\limits_{t\to\infty}f^{(p+1)}(t+\mathrm{i}\xi)=0$ locally uniformly;

  3. 3.

$\int_{-\infty}^{\infty}\mathrm{e}^{-2\pi|\xi|}|f(t+\mathrm{i}\xi)|\mathrm{d}\xi$
is locally bounded for $\mathrm{Re}t>0$,

then we have

$(\mathrm{MS})\text{-}\frac{\mathrm{d}}{\mathrm{d}z}\sum_{k=1}^{z}f(k)=f(\mathrm{B})+(\mathrm{MS})\text{-}\sum_{k=1}^{z}\frac{\mathrm{d}}{\mathrm{d}k}f(k).$

###### Proof.

Apply Theorem 4 and Lemma 2.5. ∎

In fact, the condition of $f$ in Corollary 3.1 also satisfies the third
condition in Corollary 3.2.

At this point, we justify the speculation from Müller and Schleicher. Next, we
will see that these conditions are sufficient to cover the special series they
care about.

Finally, it needs to be emphasized that the processing of Bernoulli umbrae is
essentially equivalent to applying the Abel-Plana formula. Although we have
removed the decay condition of the function when the imaginary part is large,
this is actually done by requiring the function to be exponential growth.

###  3.1 Gosper series

Let’s consider a sufficiently general example.

Suppose

$\displaystyle J(z)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{a_{2n}}{(2n)!}z^{2n}$ (3.3)

satisfies that

  1. 1.

$|a_{n}|\leq C\cdot(1+n)^{\nu_{a}}$;

  2. 2.

$|J(z)|\leq
C\cdot\mathrm{e}^{|\mathrm{Im}z|}(1+|z|)^{\nu_{J}}\quad\mathrm{Re}z>0$.

######  Lemma 3.2.

  1. 1.

$|\mathrm{Im}\sqrt{z_{1}+z_{2}}|\leq|\mathrm{Im}\sqrt{z_{1}}|+|\mathrm{Im}\sqrt{z_{2}}|$;

  2. 2.

$|\tilde{J}(z)|\leq C\cdot\mathrm{e}^{|z|}(1+|z|)^{\nu_{a}}$, where
$\tilde{J}(z)=\sum_{n=0}^{\infty}\frac{|a_{2n}|}{(2n)!}z^{2n}$.

######  Proposition 3.1.

$z^{-n}J(\sqrt{(2\pi z)^{2}+b^{2}})$ satisfies all the conditions of Corollary
3.1, where $n>\max\\{\nu_{a},\nu_{J}\\}+1$.

###### Proof.

Suppose that

$z^{-n}J(\sqrt{z^{2}+b^{2}})=\sum_{m=1}^{M}b_{m}z^{-m}+\sum_{n=0}^{\infty}\frac{c_{n}}{n!}z^{n},$

only need to notice that

$\displaystyle\int_{1}^{\infty}x^{-n}\tilde{J}(\sqrt{x^{2}+|b|^{2}})\mathrm{e}^{-x}\mathrm{d}x<\infty\Rightarrow\sum_{n=0}^{\infty}|c_{n}|<\infty,$

and for $n>\nu_{a}+1$ we have

$\displaystyle\int_{1}^{\infty}x^{-n}\tilde{J}(\sqrt{x^{2}+|b|^{2}})\mathrm{e}^{-x}\mathrm{d}x\leq
C\cdot\int_{1}^{\infty}x^{\nu_{a}-n}\mathrm{e}^{\sqrt{x^{2}+|b|^{2}}-x}\mathrm{d}x<\infty.$

When the real part of $z$ is large enough, we have

$\displaystyle\mathrm{e}^{-|\mathrm{Im}z|}|z^{-n}J(\sqrt{z^{2}+b^{2}})|$
$\displaystyle\leq
C\cdot\mathrm{e}^{|\mathrm{Im}\sqrt{z^{2}+b^{2}}|-|\mathrm{Im}z|}|z|^{\nu_{J}-n}$
$\displaystyle\leq C\cdot|z|^{\nu_{J}-n},$

and for $n>\nu_{J}+1$ we have
$z^{-n}J(\sqrt{z^{2}+b^{2}})\in\mathcal{T}_{-1,0;1,0}^{(-1)}$. ∎

For example, we can take

$J(z)=z^{-\nu}J_{\nu}(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n}}{2^{\nu+2n}n!(n+\nu)!}.$
(3.4)

From the Poisson integral representation, for $\mathrm{Re}\nu>-\frac{1}{2}$ we
have

$\displaystyle J(z)$
$\displaystyle=\frac{1}{2^{\nu}(\nu-\frac{1}{2})!\sqrt{\pi}}\int_{0}^{\pi}\mathrm{e}^{\mathrm{i}z\cos\theta}\sin^{2\nu}\theta\mathrm{d}\theta$
$\displaystyle|J(z)|$ $\displaystyle\leq C\cdot\mathrm{e}^{|\mathrm{Im}z|},$

therefore $\nu_{a}=-(\mathrm{Re}\nu+\frac{1}{2}),\nu_{J}=0$. In particular,
when $\nu=1/2$ we have $J(z)=\sqrt{\frac{2}{\pi}}\frac{\sin z}{z},n>0$, when
$\nu=-1/2$ we have $J(z)=\sqrt{\frac{2}{\pi}}\cos z,n>1$.

We can improve the related results in [5], that is, generalize $b$ to every
complex number.

######  Corollary 3.3 (Gosper series).

If $J(z)=\sum_{n=0}^{\infty}\frac{a_{2n}}{(2n)!}z^{2n}$ satisfies that

  1. 1.

$|a_{n}|\leq C\cdot(1+n)^{\nu_{a}}$;

  2. 2.

$|J(z)|\leq
C\cdot\mathrm{e}^{|\mathrm{Im}z|}(1+|z|)^{\nu_{J}}\quad\mathrm{Re}z>0$,

and $b\in\mathbb{C}$, then

  1. 1.

For $\nu_{a},\nu_{J}<0$ we have

$\displaystyle(\mathrm{MS})\text{-}\sum_{n=1/4}^{-1/4}\frac{J(\sqrt{b^{2}+(2\pi
n)^{2}})}{n}$ $\displaystyle=\pi J(b)$ (3.5)
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n+\frac{1}{2}}J(\sqrt{b^{2}+\pi^{2}(n+1/2)^{2}})$
$\displaystyle=\frac{\pi}{2}J(b);$ (3.6)

  2. 2.

For $\nu_{a},\nu_{J}<1$ we have

$\displaystyle(\mathrm{MS})\text{-}\sum_{n=1}^{-1/2}\frac{J(\sqrt{b^{2}+(2\pi
n)^{2}})}{n^{2}}$
$\displaystyle=-\frac{\pi^{2}J(b)}{3}-\frac{\pi^{2}J^{\prime}(b)}{b}$ (3.7)
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n^{2}}J(\sqrt{b^{2}+\pi^{2}n^{2}})$
$\displaystyle=-\frac{\pi^{2}J(b)}{12}-\frac{\pi^{2}J^{\prime}(b)}{4b}.$ (3.8)

###### Proof.

By Proposition 3.1, we can apply Corollary 3.1 on $z^{-n}J(\sqrt{(2\pi
z)^{2}+b^{2}})$. ∎

By applying $|f(a)-f(b)|\leq M|a-b|$, we are able to generalize further to the
situation where $\sqrt{z^{2}+b^{2}}$ is replaced by $\sqrt[n]{z^{n}+p(z)}$.

The common processing on $\sum_{n\in\mathbb{Z}}(-1)^{n}f(n)$ or
$\sum_{n\in\mathbb{Z}}f(n)$ is to apply the Abel-Plana formula, which is
essentially equivalent to the processing here.

###  3.2 Bernoulli umbrae

This subsection will mainly introduce how to use Bernoulli umbrae $\mathrm{B}$
to synthesize some common analysis results. This approach can also be easily
extended to similar umbrae such as Euler umbrae $\mathrm{E}$.

The proof of the following theorem can actually be given directly by its
corollaries, so it is omitted here.

######  Theorem 10 (Bernoulli umbrae).

  1. 1.

$\ln\mathrm{B}=-\gamma$;

  2. 2.

$\mathrm{B}\ln\mathrm{B}=\ln\sqrt{\frac{\mathrm{e}}{2\pi}}$;

  3. 3.

$\mathrm{B}^{2}\ln\mathrm{B}=\frac{1}{4}-2\ln A$, where $A$ is
Glaisher–Kinkelin constant;

  4. 4.

$\mathrm{B}^{z}=-z\zeta(1-z)$.

######  Lemma 3.3 (Multiplication theorem).

$n\mathrm{B}[+](n\mathrm{B}-1)[+]\cdots[+](n\mathrm{B}-n+1)=[n]+\mathrm{B}.$

In fact, this lemma synthesized all the multiplication theorems about special
functions.

######  Lemma 3.4.

$\left(\frac{\mathrm{d}}{\mathrm{d}z}\right)^{n}\mathrm{B}^{z}=\mathrm{B}^{z}\ln^{n}\mathrm{B}$.

###### Proof.

Apply the argument similar to Theorem 4, note that

$\int_{-\infty-\mathrm{i}t}^{\infty-\mathrm{i}t}|\hat{\mathcal{A}}(z)|\cdot|(\mathrm{i}z)^{z_{0}}\ln^{n}(\mathrm{i}z)|\mathrm{d}z$

is locally uniformly bounded for $z_{0}$. ∎

The previous definitions and conclusions such as Theorem 4, 5, 7 have directly
justified the validity of each step of the following proofs.

######  Corollary 3.4 ($\ln\mathrm{B}$).

  1. 1.

$\lim\limits_{n\to\infty}\left(\sum_{k=1}^{n}\frac{1}{n}-\ln n\right)=\gamma$;

  2. 2.

$\zeta(1+s)=\frac{1}{s}-\gamma+o(1)$;

  3. 3.

$(x!)^{\prime}(0)=-\gamma$.

###### Proof.

$\displaystyle\sum_{k=1}^{n}\frac{1}{n}-\ln n$
$\displaystyle=\ln(\mathrm{B}+n)+\gamma-\ln n$
$\displaystyle=\gamma+\ln(1+\frac{\mathrm{B}}{n})\to\gamma$
$\displaystyle\zeta(1+s)$
$\displaystyle=\frac{\mathrm{B}^{-s}}{s}=\frac{1-s\ln\mathrm{B}+o(s)}{s}$
$\displaystyle=\frac{1}{s}-\gamma+o(1)$
$\displaystyle\frac{(x!)^{\prime}}{x!}$ $\displaystyle=\ln(\mathrm{B}+x)$
$\displaystyle(x!)^{\prime}(0)$ $\displaystyle=0!\ln\mathrm{B}=-\gamma.$

∎

######  Corollary 3.5 ($\mathrm{B}\ln\mathrm{B}$).

  1. 1.

$n!\sim(\frac{n}{\mathrm{e}})^{n}\sqrt{2\pi n}$;

  2. 2.

$\zeta(s)=-\frac{1}{2}-s\ln\sqrt{2\pi}+o(s)$;

  3. 3.

$(-\frac{1}{2})!=-\frac{\sqrt{\pi}}{2}$.

###### Proof.

$\displaystyle\ln n!$
$\displaystyle=(\mathrm{B}+n)\ln(\mathrm{B}+n)-n-\ln\sqrt{\frac{\mathrm{e}}{2\pi}}$
$\displaystyle=(\mathrm{B}+n)\ln
n+(\mathrm{B}+n)\ln(1+\frac{\mathrm{B}}{n})-n-\ln\sqrt{\frac{\mathrm{e}}{2\pi}}$
$\displaystyle=(\frac{1}{2}+n)\ln
n+\mathrm{B}^{1}+O(\frac{1}{n})-n-\ln\sqrt{\frac{\mathrm{e}}{2\pi}}$
$\displaystyle=(\frac{1}{2}+n)\ln
n-n+(\frac{1}{2}-\ln\sqrt{\frac{\mathrm{e}}{2\pi}})+O(\frac{1}{n})$
$\displaystyle\zeta(s)$
$\displaystyle=\frac{\mathrm{B}^{-s+1}}{s-1}=-\mathrm{B}^{1}+s(\mathrm{B}^{1}-\mathrm{B}\ln\mathrm{B})+o(s)$
$\displaystyle=-\frac{1}{2}-s\ln\sqrt{2\pi}+o(s)$
$\displaystyle\ln(-\frac{1}{2})!$
$\displaystyle=\frac{2\mathrm{B}-1}{2}\ln\frac{2\mathrm{B}-1}{2}-\frac{1}{2}-\mathrm{B}\ln\mathrm{B}$
$\displaystyle(\text{by Theorem \ref{naturalthem}, Lemma
\ref{Multiplicationtheorem}})$
$\displaystyle=(2\cdot\frac{\mathrm{B}}{2}\ln\frac{\mathrm{B}}{2}-\mathrm{B}\ln\mathrm{B})-\frac{1}{2}-\mathrm{B}\ln\mathrm{B}$
$\displaystyle=-\mathrm{B}\ln 2-\frac{1}{2}-\mathrm{B}\ln\mathrm{B}$
$\displaystyle=-\frac{1}{2}\ln
2-\frac{1}{2}-\ln\sqrt{\frac{\mathrm{e}}{2\pi}}=\ln\sqrt{\pi}.$

∎

##  4 Some possible developments

If we only apply the Bernoulli umbrae, then what we are actually doing is
using that notation to simplify the application of the Abel-Plana formula. So
the key points seem to be on the calculus between these umbrae.

For $A_{i}=(\mathcal{A}_{i}(z),\Omega_{a,b})$, we can define

$A_{1}\times
A_{2}\coloneqq(\mathrm{e}^{\frac{1}{z}\ln\frac{\mathcal{A}_{1}(z)}{\mathcal{A}_{1}(0)}\ln\frac{\mathcal{A}_{2}(z)}{\mathcal{A}_{2}(0)}},\Omega_{a,b})$

where $A_{1}$ and $A_{2}$ are both invertible respect to the addition. It is
again a umbrae, and gives a rare three-level arithmetic, i.e. “$\times$”
distributes over “$+$”, and “$+$” distributes over “$[+]$”.

Let $\mu=(\mu_{n})_{n\in\mathbb{N^{+}}}$ be a given sequence. Suppose
$\sum_{n=1}^{\infty}\mu_{n}z^{n}$ converges near the origin, such that
$\mathcal{A}(z)=z^{m}\sum_{n=1}^{\infty}\mu_{n}\mathrm{e}^{nz}$ can be
continuation to $\Omega_{a,b}$. Denote

$(m\times\mathrm{D})+\mathrm{S}(\mu)\coloneqq(\mathcal{A}(z),\Omega_{a,b}).$
(4.1)

We can expect that for a suitable $f$, such as
$f\in\mathcal{T}_{a,k;b,l}^{(p-1)}$, we have

$\sum_{n=1}^{\infty}\mu_{n}f(n)=f^{(-m)}(\mathrm{S}(\mu)).$ (4.2)

This type of umbrae allows us to promote our previous treatment on Bernoulli
umbrae further. For example,

  1. 1.

$(m\times\mathrm{D})+\mathrm{S}(\mu)=\mathrm{B}$, where $m=1$ and
$\mu_{n}=-1$;

  2. 2.

$(m\times\mathrm{D})+\mathrm{S}(\mu)=\mathrm{E}$, where $m=0$ and
$\mu_{n}=2\chi_{1}(n)$ where $\chi_{1}$ is the non-trivial Dirichlet character
of modulo 4.

Unfortunately, not like the Dirichlet character, the sequence $\mu$ seems
cannot be replaced with a number theory sequence that is too irregular. For
example, $\sum_{n=1}^{\infty}z^{n^{2}}$ is full of singularities on the entire
unit circle.

Through the symmetry of the Fourier transformation, we are able to calculate
$A^{-n}$ more directly, which is essentially the Ramanujan’s master theorem,
i.e.

$\int_{0}^{\infty}\mathrm{e}^{-Ax}x^{n}\mathrm{d}x=n!A^{-n-1}.$ (4.3)

It seems there is a chance to achieve more similar results.

We can use Bernoulli umbrae to deal with some series in [1]. Although the
process is relatively complicated, the approach is very straightforward.

######  Lemma 4.1.

For suitable $f$, we have

$\displaystyle\sum_{n=1}^{z}\left(n\sum_{k=1}^{n}f(k)\right)$
$\displaystyle=-f^{(-3)}(\mathrm{B}+\mathrm{B}+z)+zf^{(-2)}(\mathrm{B}+\mathrm{B}+z)+f^{(-2)}(\mathrm{B}+\tilde{\mathrm{B}}+z)$
$\displaystyle\phantom{=}-\frac{z^{2}+z}{2}f^{(-1)}(\mathrm{B})-f^{(-2)}(\mathrm{B}+\tilde{\mathrm{B}})+f^{(-3)}(\mathrm{B}+\mathrm{B}),$

where $\tilde{\mathrm{B}}^{n}=\mathrm{B}_{n+1}$, and the two $\mathrm{B}$s in
$\mathrm{B}+\mathrm{B}$ should be understood as different umbraes.

######  Lemma 4.2.

Suppose
$\tilde{\mathrm{B}}^{n}=\mathrm{B}_{n+1},\tilde{\tilde{\mathrm{B}}}^{n}=\mathrm{B}_{n+2}$.

  1. 1.

For suitable $f$, we have
$f(\tilde{\mathrm{B}})=\mathrm{B}f(\mathrm{B}),f(\tilde{\tilde{\mathrm{B}}})=\mathrm{B}^{2}f(\mathrm{B})$;

  2. 2.

$\mathrm{B}+\mathrm{B}=\mathrm{B}[+](\mathrm{B}+\mathrm{D})[-](\tilde{\mathrm{B}}+\mathrm{D})$;

  3. 3.

$\mathrm{B}+\tilde{\mathrm{B}}=(\mathrm{B}[+](\tilde{\mathrm{B}}+\mathrm{D})[-](\tilde{\tilde{\mathrm{B}}}+\mathrm{D}))+[\frac{1}{2}]$.

Finally, we can also see that if we allow the umbrae to be written as the
difference between two analytic functions with disjoint domains, like the
Fourier transform of the function $f$, we can eventually cover more general
integral transforms.

## References

  * [1] V. S. Adamchik.  The multiple gamma function and its application to computation of series.  Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan, 9(3):271–288, 2005.
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  * [7] Sandy Grabiner.  Using Banach algebras to do analysis with the umbral calculus.  In Conference on Automatic Continuity and Banach Algebras (Canberra, 1989), volume 21 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 170–185. Austral. Nat. Univ., Canberra, 1989.
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  * [9] Mourad E. H. Ismail and Dennis Stanton.  More orthogonal polynomials as moments.  In Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), volume 161 of Progr. Math., pages 377–396. Birkhäuser Boston, Boston, MA, 1998.
  * [10] Markus Müller and Dierk Schleicher.  How to add a non-integer number of terms, and how to produce unusual infinite summations.  Journal of Computational and Applied Mathematics, 178(1-2):347–360, 2005.
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# On the interaction of viscoelasticity and waviness in enhancing the pull-off
force in sphere/flat contacts

M. Ciavarella  Politecnico di BARI. DMMM department. Viale Gentile 182, 70126
Bari<EMAIL_ADDRESS>
###### Abstract

Motivated by roughness-induced adhesion enhancement (toughening and
strengthening) in low modulus materials, we study the detachment of a sphere
from a substrate in the presence of both viscoelastic dissipation at the
contact edge, and roughness in the form of a single axisymmetric waviness. We
show that the roughness-induced enhancement found by Guduru and coworkers for
the elastic case (i.e. at very small detachment speeds) tends to disappear
with increasing speeds, where the viscoelastic effect dominates and the
problem approaches that of a smooth sphere. This is in qualitative agreement
with the original experiments of Guduru’s group with gelatin. The cross-over
velocity is where the two separate effects are comparable. Viscoelasticity
effectively damps roughness-induced elastic instabilities, and make their
effects much less important.

###### keywords:

Roughness, Adhesion, Guduru’s theory, viscoelasticity

††journal: journal

##  1
Introduction

It is well known that adhesion of hard solids is difficult to measure at
macroscopic scales, and Fuller and Tabor (1975) proved that even in low
modulus materials (they used rubbers with $E\sim 1$ $\mathrm{MPa}$), a $\sim
1\left[\mu m\right]$ of roughness destroy adhesion almost completely, despite
van der Waals adhesive forces are quite strong, giving the so called ”adhesion
paradox” (Kendall, 1975). Adhesion of macroscopic bulk objects requires smooth
surfaces, and at least one of the solids has to have a very low elastic
modulus. Dahlquist (1969a, 1969b) while working at 3M proposed a criterion
largely used in the world of adhesives, namely that the elastic Young modulus
should be smaller than $\sim 1$ $\mathrm{MPa}$ to achieve stickiness even in
the presence of roughness. This clearly is just a rough indication, but Tiwari
et al.(2017) find for example that the work of adhesion (at a given retraction
speed) is reduced of a factor 700 for a rubber in contact with a rough hard
sphere when the rubber modulus is $E=2.3MPa$, but is actually increased
because of roughness by a factor 2 when the rubber has modulus $E=0.02MPa$.
The threshold doesn’t change much even if we consider nanometer scale
roughness, as in the recent results of Dalvi et al. (2019) for pull-off of
PDMS hemispheres having four different elastic moduli against different
roughened plates: Dahlquist’s criterion seems to work surprisingly well, as
while there is little effect of roughness for the 3 cases of low modulus up to
near $E=2\left[MPa\right]$, roughness has strong effect both during approach
and retraction for the high modulus material ($E=10\left[MPa\right])$, where
the hysteresis may be due partly to viscoelastic effects111Despite the authors
intended to remove as much as possible rate-dependent effects by applying a
retraction rate of only $60nm/s$). The authors claim a good correlation of the
energy loss during the cycle of loading and withdrawing with the product of
the real contact area at maximum preload with the ”intrinsic” work of
adhesion. Notice that this would not work for a smooth sphere where JKR theory
predicts that the energy loss is independent on preload and indeed the data of
Dalvi et al. (2019) with the lowest roughness do show almost a constant trend.
Also, the hard material case shows almost no energy loss.. However, for the 3
low modulus materials, roughness almost systematically increases the work of
adhesion rather than decreasing it as for the high modulus material, for a
given retraction speed.

Roughness-induced adhesion enhancement was measured with some surprise first
by Briggs & Briscoe (1977), and Fuller & Roberts (1981), and Persson-Tosatti’s
(2001) theory above attributes it to the increase of surface area induced by
roughness222Whereas adhesion reduction is attributed by Persson and Tosatti
(2001) to the elastic energy to flatten roughness, which is proportional to
the elastic modulus.. Another mechanism was put forward by Guduru and
collaborators (Guduru, 2007, Guduru and Bull, 2007). Guduru considered a
spherical contact having a concentric axisymmetric waviness, and considers the
contact is complete over the contact area. The waviness gives rise to
oscillations in the load-approach curve which result in up to factor 20
increase of the pull-off with respect to the standard smooth sphere case of
the JKR theory (Johnson, Kendall & Roberts, 1971). Also, the curves fold on
each other so that we expect jumps at some points in the equilibrium curve.
which corresponds to dissipation and emission of elastic waves in the
material, and results in strong hysteresis. Later, Kesari and Lew (2011)
noticed that Guduru’s solution has an elegant ”envelope” obtained by expanding
asymptotically for very small wavelength of the waviness.

But most soft materials are viscoelastic, and therefore there is a strong
velocity dependence of the pull-off result. Many authors (Gent and Schultz,
1972, Barquins and Maugis 1981, Gent, 1996, Gent & Petrich 1969, Andrews &
Kinloch, 1974, Barber et al, 1989, Greenwood & Johnson, 1981, Maugis &
Barquins, 1980, Persson & Brener, 2005) have proposed that the process of
peeling involves an effective work of adhesion $w$ which is the product of the
thermodynamic (Dupré) work of adhesion $w_{0}$ and a function of velocity of
peeling of the contact/crack line and temperature, as long as there is no bulk
viscoelasticity involved, over a large range of crack speeds, namely of the
form that has been validated also by a large amount of data including peeling
tests at various peel angles

$w=w_{0}\left[1+k\left(a_{T}v_{p}\right)^{n}\right]$ (1)

where $k,n$ are constants of the material, with $n$ in the range $0.1-0.8$ and
$a_{T}$ is the WLF factor (Williams, Landel & Ferry, 1955) which permits to
translate results at various temperatures $T$ from measurement at a certain
standard temperature. This form of effective work of adhesion was obtained
also from theoretical models using either Barenblatt models or crack tip
blunting models (Barber Donley and Langer 1989, Greenwood and Johnson, 1981,
Persson & Brener, 2005) generally using simple a single relaxation time model
for the materials. Actually, Persson & Brener (2005) showed that for a
frequency dependent viscoelastic modulus
$E\left(\omega\right)\sim\omega^{1-s}$, $0<s<1$, in the transition region
between the ”rubbery region” and the ”glassy region” (where the strong
internal damping occurs important for energy loss processes), the equation (1)
is satisfied at intermediate velocities with
$n=\left(1-s\right)/\left(2-s\right)$ (so that $0<n<1/2,$ in agreement with
most of the range cited above). There may be deviations for materials having a
complex behaviour with many relaxation times or for considering the role of
finite size of the system or of very high temperature at the crack tip which
depends on thermal diffusivity of the material, see recent review (Rodriguez
et al., 2020).

The effective ”toughness” $w$ can increase of various orders of magnitude over
$w_{0}$ as the velocity increases (more precisely, of the ratio
$E\left(\infty\right)/E\left(0\right)$, where $E\left(\omega\right)$ is the
frequency dependent elastic modulus), and the pull-off of a sphere has also
been effectively measured to increase of various orders of magnitude over an
increase of peeling speed (Barquins & Maugis, 1981). On the contrary, during
crack closure the effective work of adhesion is even smaller than $w_{0}$,
(this time it is reduced by the ratio $E\left(0\right)/E\left(\infty\right)$,
see Greenwood and Johnson, 1981), so in some cases loading could become
essentially an elastic model without adhesion.

Equation (1) generalizes the thermodynamic equilibrium of elastic cracks for
the strain energy release $G$: namely, it provides a condition for crack edge
velocity — when $G>w_{0}$, the crack accelerate under the force $G-w_{0}$
applied per unit length of crack, until a limit speed $v_{p}$ for equilibrium
is found, depending on the loading conditions. For example, $G-w_{0}$ is a
constant for classical peeling experiments, whereas it monotonically increases
for flat punches, and has a much richer behaviour for the smooth sphere.
Therefore, for imposed tensile load smaller in absolute value than the JKR
pull-off value $P_{0}=3/2\pi wR$, the contact area simply decreases to another
equilibrium value (given asymptotically by JKR theory), while for imposed load
below the JKR value, it decreases with non monotonic velocity but without the
JKR pull-off instability, so up to complete detachment. Therefore, pull-off
depends on the loading condition: can be anything greater than $P_{0}$ if load
is imposed, whereas it is a precise function of the retraction rate in an
experiment where the cross-head of a rigid machine keeps the remote approach
velocity as constant.

Various authors (Barquins & Maugis, 1981, Greenwood & Johnson, 1981, Muller,
1999) have studied the peeling of viscoelastic spheres with the above form of
fracture mechanics formulation (1), and some approximate scaling results have
also been given (Muller, 1999), but a theoretical or numerical investigation
about the coupled effect of viscoelasticity and roughness has not been
attempted, in the best of the author’s knowledge, not even with numerical
simulations. It seems that in general viscoelasticity can only increase the
”tack” i.e. the force or the work needed to detach two solids, whereas the
role of roughness is more controversial, as we have discussed above. We are
aware of the complexity of the general problem, so here, we tackle the study
of a simple problem, that of a sphere with a single wave of roughness, which
generalizes the relatively recent work of Guduru and collaborators (Guduru,
2007, Guduru and Bull, 2007) and following related literature, to the case of
a viscoelastic substrate.

##  2 The theory

We consider the Guduru contact problem for a sphere against a flat surface,
where the gap is defined as
$f\left(r\right)=\frac{r^{2}}{2R}+A\left(1-\cos\frac{2\pi r}{\lambda}\right)$,
where $R$ is the sphere radius, $\lambda$ is wavelength of roughness, $A$ is
its amplitude.

---

Fig.1 - The geometry of the problem. A sphere of radius $R$ with a simple
roughness being a single axisymmetric wave with wavelength $\lambda$ and
amplitude $A$.

The Guduru problem can be solved by considering the stress intensity factor
$K$ at the contact edge (radius $r=a$) or equivalently the strain energy
release rate $G$ (Guduru, 2007)

$G\left(a,P\right)=\frac{K\left(a,P\right)^{2}}{2E^{\ast}}=\frac{\left(P_{1}\left(a\right)-P\right)^{2}}{8\pi
E^{\ast}a^{3}}$ (2)

where $E^{\ast}=E/\left(1-\nu^{2}\right)$ is plane strain elastic modulus
(i.e. $E$ is Young’s modulus and $\nu=0.5$ Poisson ratio generally equal to
0.5 in rubbery materials, while we consider the countersurface is generally
much more rigid so we neglect its elastic properties).

Here, $P_{1}\left(a\right)$ is the load required to maintain a contact radius
$a$ in the absence of adhesion, while $P$ is the smaller load to maintain the
same contact radius in the presence of adhesion. In particular standard
contact mechanics gives (Guduru, 2007)

$P_{1}\left(a\right)=2E^{\ast}\left\\{\left(\frac{2}{R}+\frac{4\pi^{2}A}{\lambda^{2}}\right)\frac{a^{3}}{3}+\frac{\pi
Aa}{2}H_{1}\left(\frac{2\pi
a}{\lambda}\right)-\frac{\pi^{2}Aa^{2}}{\lambda}H_{2}\left(\frac{2\pi
a}{\lambda}\right)\right\\}$ (3)

where $H_{n}$ are Struve functions of order $n$.

In the adhesionless conditions, the remote approach (positive for compression)
is

$\alpha_{1}\left(a\right)=\frac{a^{2}}{R}+\pi^{2}\frac{A}{\lambda}aH_{0}\left(\frac{2\pi
a}{\lambda}\right)$ (4)

so in the adhesive condition we have to decrease this by an amount given by a
flat punch displacement giving the general result for approach

$\alpha\left(a,P\right)=\frac{a^{2}}{R}+\pi^{2}\frac{A}{\lambda}aH_{0}\left(\frac{2\pi
a}{\lambda}\right)-\frac{P_{1}\left(a\right)-P}{2E^{\ast}a}$ (5)

From (5), we can obtain the general equation for the load as a function of
contact radius and approach

$P\left(a,\alpha\right)=P_{1}\left(a\right)+2E^{\ast}a\alpha\left(a\right)-2E^{\ast}\frac{a^{3}}{R}-\pi^{2}\frac{2E^{\ast}A}{\lambda}a^{2}H_{0}\left(\frac{2\pi
a}{\lambda}\right)$ (6)

where $P_{1}\left(a\right)$ is given by (3) above. Imposing the condition of
thermodynamic equilibrium $G\left(a\right)=w_{0}$ using (2) and (3) permits to
write the Guduru solution explicitely as parametric equations of the contact
radius $a$

$\displaystyle P\left(a\right)$
$\displaystyle=P_{1}\left(a\right)-a^{3/2}\sqrt{8\pi w_{0}E^{\ast}}$ (7)
$\displaystyle\alpha\left(a\right)$
$\displaystyle=\alpha_{1}\left(a\right)-a^{1/2}\sqrt{2\pi w_{0}/E^{\ast}}$ (8)

Using the Kesari & Lew (2011) expansion, Ciavarella (2016) obtained that the
Guduru solution has oscillations bounded between two exact JKR (Johnson,
Kendall & Roberts, 1971) envelope curves for the smooth sphere, but with a
corrected (enhanced or reduced, respectively for unloading or loading) surface
energy

$\displaystyle P_{env}\left(a\right)$
$\displaystyle=\frac{4}{3R}E^{\ast}a^{3}-a^{3/2}\sqrt{8\pi
wE^{\ast}}\left(1\pm\frac{1}{\sqrt{\pi}\alpha_{KLJ}}\right)$ (9)
$\displaystyle\alpha_{env}\left(a\right)$
$\displaystyle=\frac{a^{2}}{R}-a^{1/2}\sqrt{\frac{2\pi
w}{E^{\ast}}}\left(1\pm\frac{1}{\sqrt{\pi}\alpha_{KLJ}}\right)$ (10)

where

$\alpha_{KLJ}=\sqrt{\frac{2w_{0}\lambda}{\pi^{2}E^{\ast}A^{2}}}$ (11)

is the parameter Johnson (1995) introduced for the JKR adhesion of a nominally
flat contact having a single scale sinusoidal waviness of amplitude $A$ and
wavelength $\lambda$. Thus, since (9,10) are JKR equation for a smooth sphere
of radius $R$, the factor

$\frac{w_{eff}}{w_{0}}=\left(1+\frac{1}{\sqrt{\pi}\alpha_{KLJ}}\right)^{2}$
(12)

is an roughness-induced increase which holds as long as a compact contact area
can be obtained, which requires not too large roughness, and/or sufficiently
strong precompression. In practice, factors up to 20 have been obtained also
experimentally by Guduru & Bull (2007), although of course these were achieved
in geometry built for the specific goal to achieve very large enhancement.
Fig.2 elucidates the behaviour of the oscillations in the Guduru solution for
a representative case, which we shall later extend to the viscoelastic
solution. Given these gulfs and reentrances, in the elastic solution, the real
followed path will depend on the loading condition. For a soft system (close
to ”load control”), there will be horizontal jumps in approach while in a
stiff system (close to ”displacement control”) there will be vertical jumps to
the next available stable position. In both cases there will be areas
”neglected” during these jumps which represent mechanical dissipated energy.
Indeed, in the ”envelope” solution of Kesari-Lew-Ciavarella, the combined
effect of these jumps results in the different JKR loading and unloading
curves which give an additional hysteresis with respect to the standard JKR
case, where the only hysteresis comes a single elastic instability in pull-in
and another (different) single instability at pull-off. The dashed lines in
Fig.2 are the Kesari-Lew-Ciavarella envelopes (9,10) using (11).

---

Fig.2 - The load-approach curve in the Guduru elastic problem with
$E^{\ast}=16500\left[Pa\right]$; $R=0.23\left[m\right]$;
$w_{0}=0.008\left[J/m^{2}\right]$; $\lambda/R=0.002$; $A/\lambda=0.005$, and
reference $a_{i}$ $=0.01\left[m\right]$.

###  2.1
Viscoelastic problem

For a given remote applied withdrawing of the sphere $v=-\frac{d\alpha}{dt}$,
we can write the velocity of the contact edge as

$v_{p}=-\frac{da}{dt}=v\frac{da}{d\alpha}$ (13)

The condition $G\left(a\right)=w$ (which replaced the thermodynamic
equilibrium $G\left(a\right)=w_{0}$ for the elastic sphere) therefore defines
a differential equation for $a=a\left(\alpha\right)$, obtained using (2,1,13)

$\frac{1}{k^{1/n}a_{T}v}\left(\frac{\left(P_{1}\left(a\right)-P\right)^{2}}{8\pi
E^{\ast}a^{3}w_{0}}-1\right)^{1/n}=\frac{da}{d\alpha}$ (14)

Hence, using (6) and defining dimensionless parameters

$V=k^{1/n}a_{T}v\qquad;\qquad\zeta=\left(\frac{2\pi
w_{0}}{RE^{\ast}}\right)^{1/3}$ (15)

we write (14) as

$\frac{da}{d\alpha}=\frac{1}{V}\left[\frac{\left(R/a\right)}{\zeta^{3}}\left(\frac{\alpha}{R}-\frac{a^{2}}{R^{2}}-\pi^{2}\frac{A}{\lambda}\frac{a}{R}H_{0}\left(\frac{2\pi
a}{\lambda}\right)\right)^{2}-1\right]^{1/n}$ (16)

which can be solved with a numerical method. After a solution is obtained for
$a=a\left(\alpha\right)$, we substitute back into (6) to compute the load.
Notice that, for a given starting point of the peeling process in terms of
load $P$, the term under parenthesis in (14) is zero, and hence
$\frac{da}{d\alpha}$ starts off zero giving some delay with respect to the
elastic curve, which is hard to eliminate even at very low withdrawal speeds.

##  3 Results

###  3.1 Rough sphere

We consider first a viscoelastic material having $n=0.33$, and dimensionless
withdrawal velocity $V=0.0002,0.002,0.02,0.2,2$; the other constants, as
indicated in Fig.1, are $E^{\ast}=16500\left[Pa\right]$;
$R=0.23\left[m\right]$; $w_{0}=0.008\left[J/m^{2}\right]$; $\lambda/R=0.002$;
$A/\lambda=0.005$. This corresponds to an ”adiabatic” elastic enhancement of
the pull off according to the equation derived from the Guduru theory (12) of
$w_{eff}/w_{0}=1.42$. We indicate with $P_{0}=3/2\pi w_{0}R$ the JKR value of
pull-off for the smooth sphere. The loading curve follows the elastic solution
(7, 8), and we start withdrawing the indenter from a reference value of
$a_{i}$ $=0.01\left[m\right]$. Corresponding values of initial approach
$\alpha_{i}$ and load $P_{i}$ can therefore found from (7, 8). Numerical
solutions are found with the NDSolve algorithm in Mathematica with default
options. Fig.3a shows the obtained load-approach curve in terms of $P/P_{0}$
and $\alpha/\alpha_{i}$ where $P_{0}$ is JKR pull-off of the smooth sphere,
and Fig.3b shows the contact radius $a/a_{i}$ peeling as a function of
approach. The inner black wavy curves are the equilibrium Guduru solutions,
and the other 5 curves are obtained numerically for increasing dimensionless
velocities of withdrawal $V=0.0002,0.002,0.02,0.2,2$. As expected, the
viscoelastic peeling terminates only when contact radius is zero, and not at
the JKR unstable radius. However, the minimum load is found for a contact
radius which, for low velocities, is not too different from the unstable pull-
off contact radius in JKR theory.

(a)
---
(b)
---

Fig.3 - Load $P/P_{0}$ (a) and the contact radius $a/a_{i}$ (b) as a function
of approach $\alpha/\alpha_{i}$. $P_{0}$ is JKR pull-off of the smooth sphere,
and $\alpha_{i}$ the initial value of approach for unloading. The inner black
wavy curve is the equilibrium Guduru solution, and the other 5 curves
increasingly departing from it are obtained numerically for increasing
dimensionless velocities of withdrawal (see the verse of the arrow)
$V=0.0002,0.002,0.02,0.2,2$. Here, $n=0.33$, and other constants as indicated
in Fig.1.

Fig.4a,b,c give some detail of the solution at the lowest dimensionless
velocity of withdrawal $V=0.0002$. In particular, Fig.4a shows clearly that
the numerical solution follows closely the prediction of the Guduru elastic
solution under displacement control, as expected, with almost sharp jumps of
the force at specific values of the approach. After the jump, the solution
seems to return to the Guduru equilibrium solution. Obviously with the
viscoelastic theory, the strict elastic solution should be obtained
asymptotically at extremely low velocities, but the differential equation
would then become very ”stiff” corresponding to numerical difficulties
following the jumps. The same behaviour is clarified in terms of the contact
radius in Fig.4b which follows very closely the Guduru solution in some
periods of time, then extends a little before jumping almost abruptly to the
following branch of the equilibrium curve. In other words, the curve does not
have a ”rainflow” type of behaviour over the Guduru equilibrium solution,
which would be the elastic real behaviour with jump-instabilities, but the
contact radius ”drops” over the Guduru curve only after some delays. This is
further clarified in Fig.4c, where the velocity of the contact line
$da/d\alpha=v_{p}/v$ is found to follow an oscillatory trend with ”bursts” of
very high (but finite) velocity where the peeling velocity is a much larger
value than the imposed withdrawing velocity, after which the velocity drops to
a low value which is where the contact area approaches the adiabatic Guduru
curve since $G\simeq w_{0}$, and which increases progressively with the
decreasing approach. Slowly, the solution departs from the Guduru elastic one,
because of the cumulative effects of the acceleration periods. However, from
Fig.3, and Fig.4d, for high velocities, we see that there are no real ”jumps”,
and the solution curve is generally smoother, with the difference between the
slow regime and the fast regime being smaller. Also notice in particular that
while the velocity of peeling remains in every case equal to zero at the
initial point, it remains closer to zero for a much extensive range of
approach for high velocities, resulting in a curve departing away from the
equilibrium Guduru curve immediately. This effect at high velocity produces
curves that are generally closer to the viscoelastic curves for the smooth
sphere, and therefore closer results for pull-off and work for pull-off. To
see this from a quantitative point of view, since even the smooth sphere
problem requires a numerical solution, we describe some results in the next
paragraph.

$\begin{array}[c]{cc}\begin{tabular}[c]{l}\@add@centering\centering\includegraphics[height=128.0374pt]{4a.eps}\end{tabular}&\begin{tabular}[c]{l}\<EMAIL_ADDRESS>\text{(a)}&\text{(b)}\end{array}$

$\begin{array}[c]{cc}\begin{tabular}[c]{l}\@add@centering\centering\includegraphics[height=128.0374pt]{4c.eps}\end{tabular}&\begin{tabular}[c]{l}\<EMAIL_ADDRESS>\text{(c)}&\text{(d)}\end{array}$

Fig.4 - Detail of the solution at the lowest dimensionless velocity of
withdrawal (a,b,c) $V=0.0002,$ and (d) $V=2$. Here, $n=0.33$, and other
constants as indicated in Fig.1. In particular (a) Load-approach (b) contact
radius vs approach (c) velocity of contact line $da/d\alpha=v_{p}/v$. (d)
velocity of contact line $da/d\alpha=v_{p}/v$ but for the highest
dimensionless speed $V=2$.

###  3.2 Smooth sphere

We solve the problem for the smooth sphere under the same set of conditions
(the equation for the smooth sphere are obviously obtained for $A=0$ above),
and obtained smooth results are shown in Fig.5. Notice that as we discussed in
the theory paragraph, in the initial point $\frac{da}{d\alpha}=0$, and we find
again (Fig.5c) that the velocity remains practically zero for a longer time
when $V$ is bigger. Here, contrary to the rough sphere, but also contrary to
the smooth sphere in the case when load is constant, the velocity of the
contact line increases monotonically from zero to infinite when pull-off
occurs at zero contact area.

(a)
---
(b)
---
(c)
---

Fig.5 - The load $P/P_{0}$ (a), the contact radius $a/a_{i}$ (b), and the
velocity of contact line $da/d\alpha=v_{p}/v$ (c) as a function of approach
$\alpha/\alpha_{i}$ for the smooth sphere. The inner black curve is the JKR
classical solution, and the other 5 curves are obtained numerically for
$V=0.0002,0.002,0.02,0.2,2$ (follow the arrow). Here, $n=0.33$, and other
constants as indicated in Fig.1.

###  3.3 Some comparisons

Summarizing the pull-off results for $n=0.33$, but adding some solutions also
at different amplitudes of roughness $A$, we obtain the amplification factor
for pull-off with respect to the JKR value as in Fig.6. Notice initially that
the smooth sphere results tend to a power-law scaling (linear in the log-log
plot) as expected from the material law (1), after a transition from the
elastic behaviour. As it is evident from the figure, starting off at low
velocity with increasing amplitude of roughness, increases the ”elastic”
amplification according to the Guduru theory, but eventually the effect
disappears at sufficiently large peeling speeds in the viscoelastic theory. In
other words, there seems to be a ”cross-over” between the two phenomena at the
speed for which the two increases are the same. So in Fig.6, for example, for
the very ”rough” spheres of $A/\lambda=0.045$ (blue line), we see that the
Guduru enhancement is larger (7.5) that the viscoelastic one in this velocity
range, and we don’t see a significant decrease of its effect, but eventually
the smooth sphere result would be obtained for much larger speeds.

Obviously, with so many constants in the problem, it is not easy to give
comprehensive results, but further tests with a larger value of $n=0.6$ give
results which confirm our conclusion about the crossover: namely pull-off
force increases faster with the dimensionless speed factor $V$, but the Guduru
effect disappears also faster.

$\begin{array}[c]{cc}\begin{tabular}[c]{l}\@add@centering\centering\includegraphics[height=128.0374pt]{6a.eps}\end{tabular}&\begin{tabular}[c]{l}\<EMAIL_ADDRESS>\text{(a)}&\text{(b)}\end{array}$

Fig.6 - The pull-off amplification with respect to the JKR value, $P/P_{0}$ as
a function of dimensionless speed of withdrawal $V$ for various amplitude of
waviness increasing as indicated by arrow: $A/\lambda=0$ for the smooth sphere
(black), $A/\lambda=0.005$ (green)$,A/\lambda=0.015$ (red), $A/\lambda=0.045$
(blue). Here, all constants as indicated in Fig.1, except (a) $n=0.33$; (b)
$n=0.6$.

##  4 Discussion

Considering the experiments on our geometry done by Guduru & Bull (2007),
their gelatin material is indeed a viscoelastic material, as recognized by
Guduru & Bull (2007) who however did not characterized the material in
particular and tried to minimize the loading rate effects by keeping in their
tensile test machine a crosshead velocity at $v=3mm/min=50\mu m/s$ in all
experiments. Indeed, even the smooth sphere case they use for measuring the
baseline work of adhesion shows significant deviations between loading and the
unloading curves which are not present in JKR theory. The unloading has
specific features similar to what we found for the smooth sphere, namely when
unloading begins, the contact radius does not begin to decrease immediately.
Fitting JKR curves, Guduru & Bull (2007) extracted $w_{0}=0.008N/m$ during
loading and $w_{0}^{\prime}=0.22N/m$, a difference of a factor
$27.\,\allowbreak 5$ but they used as baseline for their comparison with the
wavy surfaces the unloading value. Despite experimental results capture
generally the trend of the predictions, there is a ”systematic difference
between the experimental observation and the theoretical prediction” as the
authors say, of the order of a $-25\%$. We attribute this to the effect
observed in the present paper, namely that since Guduru & Bull (2007) used
their elastic theory with the work of adhesion already increased by
viscoelastic effects, they overestimate the effect of load amplification.

Consider next the experiments by Barquins & Maugis (1981), where material was
characterized: the viscoelastic toughness (1) was measured with $n=0.6$ in the
range $v_{p}=10^{-1}-10^{3}\mu m/s$, i.e. over 4 decades of speed, which
corresponds to an increase of 2 orders of magnitude in
$\frac{w-w_{0}}{w_{0}}$333Measuring $v_{p}\left[\mu m/s\right]$, we have
$\frac{w-w_{0}}{w_{0}}\simeq 10v_{p}^{0.6}$from $w\simeq 3.5w_{0}$ to $w\simeq
632w_{0}$.. In ”tack” experiment of a $2.2mm$ glass ball on the polyurethane
surface (i.e. pull-off experiments for which after a fixed contact time (5
min), at room temperature, a cross-head velocity is imposed in a tensile
machine), an adherence force more than 30 times the quasistatic value at fixed
displacement of the JKR theory, was found when $v=1\mu m/s$. By increasing the
cross-head speed by 4 orders of magnitude i.e. to $v=$1$mm/s$, pull-off
increased by a further factor of about 30. That is the viscoelastic effect can
be very large, and therefore generally much larger than the Guduru effect.
Also, Guduru effect only holds for a quite special waviness, and when the
contact ”peels” quite uniformly around a circle, and requires the initial
contact area to be compact, which poses some limits to the amplitude of
roughness (see a more general numerical solution using Lennard-Jones force-
separation law in Papangelo & Ciavarella, 2020). Further, Li et al. (2019)
numerical experiments for a rough sphere and elastic materials but with
2-dimensional wavy roughness (a generalization of the Guduru geometry which
does not require the contact to peel in axisymmetric manner), the adhesion
enhancement still persists although perhaps reduced (they found an increase of
a factor 1.7 for a Johnson parameter (11) $\alpha_{KLJ}=0.37$ which
corresponds to a Guduru 1D enhancement (12) of

$\frac{w_{eff}}{w_{0}}=\left(1+\frac{1}{\sqrt{\pi}0.37}\right)^{2}=6.37$ (17)

and therefore the 2D sinusoidal waviness already decreases the Guduru effect
significantly. Further numerical experiments in Li et al. (2019) with random
roughness suggest that this enhancement was reduced to a maximum factor of
1.2, much smaller than expected from Guduru theory because of the random
nature of roughness.

Returning finally to the experiments of Dalvi et al. (2019), the increase of
apparent work of adhesion in the smoother specimen (Polished
UltraNanoCrystalline Diamond (PUNCD)) is of a factor 2, at the retraction
speed of $60nm/s$. Given the discussed experimental findings of Barquins &
Maugis (1981), which we can compare only qualitatively as the material may be
quite different, if a factor 30 can be justified at $v=1\mu m/s$, the factor 2
increase cannot be excluded not even at the $60nm/s$ speed. Regarding the
roughness-induced enhancement, it was of a further factor less than 2, which
could be either due to area increase as from the Persson-Tosatti (2001)
theory, or from a reduced Guduru effect. But in view of the fact that the two
enhancements are of the same order, this factor 2 may be too large, according
to the present results and considering, to be attributed to Guduru’s effect,
especially given the roughness is probably of random nature, and so assuming
Li et al. (2019)’s results to be appropriate.

##  5 Conclusions

We have revisited the Guduru model for roughness-induced enhancement of
adhesion of a sphere/flat contact, adding the effect of viscoelasticity which
is expected in soft materials. The results have demonstrated that the
roughness-induced amplification of pull-off in the Guduru model, which
effectively can be modelled as an increased work of adhesion in the unloading
curve, is reduced progressively when velocity increases with respect to the
baseline smooth viscoelastic sphere. This is also in qualitative agreement
with the original experiments of Guduru and Bull (2007). A significant
reduction has already occurred at a ”cross-over” velocity for which the two
enhancement (the Guduru and the viscoelastic one) are of equal magnitude. We
may be tempted therefore to speculate that viscoelasticity effectively damps
the roughness-induced elastic instabilities, reduces roughness effects in
unloading, while its effects are concentrated in the loading phase.

##  6 Acknowledgements

MC acknowledges support from the Italian Ministry of Education, University and
Research (MIUR) under the program ”Departments of Excellence” (L.232/2016).

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