# Thermalon mediated AdS to dS phase transitions in Einstein-Gauss-Bonnet-
massive gravity

Supakchai Ponglertsakul<EMAIL_ADDRESS>Strong Gravity Group,
Department of Physics, Faculty of Science, Silpakorn University, Nakhon Pathom
73000, Thailand  Phongpichit Channuie<EMAIL_ADDRESS>College of Graduate
Studies, Walailak University, Thasala, Nakhon Si Thammarat, 80160, Thailand
School of Science, Walailak University, Thasala, Nakhon Si Thammarat, 80160,
Thailand  Daris Samart<EMAIL_ADDRESS>Khon Kaen Particle Physics and
Cosmology Theory Group (KKPaCT), Department of Physics, Faculty of Science,
Khon Kaen University, 123 Mitraphap Rd., Khon Kaen, 40002, Thailand

###### Abstract

In this work, gravitational phase transition emerging from anti de-Sitter
(AdS) to de-Sitter (dS) vacua in Einstein-Gauss-Bonnet-massive gravity (EGBMG)
is considered. We determine the location of thermalon (a static bubble
solution in Euclidean space) which exists in casually connected two regions of
the spacetime. The thermalon plays a major role in gravitational phase
transition by inducing the decay of the negative effective cosmological
constant to the positive one due to the higher-order gravity effects. The
thermodynamics phase space of the Hawking temperature and free energy is
investigated in details. We find that the free energy of the thermalon is
always negative at the maximum of Hawking temperature for all possible values
of the Gauss-Bonnet couplings. This means that the phase transition mediated
by thermalon from AdS to dS asymptotics is inevitable according to the
presence of the massive gravity. More importantly, the parameters of the
massive gravity behave similarly to the order parameters in the phase
transition instead of the Gauss-Bonnet coupling.

Bubble nucleation, Higher dimensional gravity, Gauss-Bonnet gravity, Massive
gravity, AdS/dS gravitational phase transition

##  I Introduction

The Cosmological Constant (CC) problem has been a major challenge in field
theory and cosmology for several decades. On the one hand, the CC is
interpreted as the vacuum energy density and the prediction of the standard
quantum field theory calculation gives the theoretical result up to the Planck
scale bigger than the observational value about 120 order of magnitudes [1].
This contradiction poses a serious problem and remains as the unsolved problem
in high energy physics. On the other hand, the CC can be used to describe the
accelerating expansion of the universe at the present time or the dark energy
problem [2, 3, 4]. Nowadays, the interpretation of the CC as dark energy
becomes the standard model of cosmology known as the Lambda Cold Dark Matter
(LCDM) and the LCDM model can describe most of the cosmological data. However,
according to the CC problem, we need the correct theory or better solution for
explaining the fundamental mechanism of the CC problem. This leads to a number
of the theoretical models proposed to solve the smallness of observed values
of the CC. In particular, an approach from the Beyond Standard Model (BSM) of
particle physics is received a lot of attention, for instances, supersymmetric
model, large extra dimension approach of CC, string landscape and etc. See
Refs.[5, 6, 7, 8, 9, 10, 11, 12, 13] for reviews on this subject. One of
interesting approach for describing the smallness of the CC value is the
gravitational phase transition. The phase transition arises from the bubble
nucleation of the true vacuum in the false vacuum spacetime when the bubble
pops up and expands until filling up the whole spacetime. This means that the
CC of the false vacuum is changed to the new value of the CC according to new
vacuum energy of the true vacuum. The mechanism of this gravitational phase
transition is originally proposed by Refs.[14, 15]. The phase transition for
Einstein gravity coupled to the 3-form fields is studied and the results
revealed that the CC can be dynamically changed into several values from the
infinite decay of the false vacuum and this leads to an alternative approach
to describe the CC problem [16, 17]. More interestingly, Ref.[18] has used the
same gravitational setting but hosting the black hole inside the bubble. Then
a thermalon, the instanton solution of the thermal trigger for the bubble
nucleation, plays the role as a mediator of the phase transition. Under this
consideration, the bubble with the three-form charge induces the decay of the
dS false vacuum into another smaller dS true vacuum. In order to thermally
trigger the phase transition, one needs the source of the exotic matter fields
with negative pressure. Alternatively, Refs.[19, 20] have proposed new
mechanism analogous to the thermalon mediated the phase transition by
introducing the higher order gravity into the system. There are two solution
branches of the higher order gravity corresponding to negatively and
positively effective cosmological constants (AdS and dS spacetime,
respectively). Initially, the false vacuum is displayed by the AdS spacetime.
The true vaccum is represented by the dS geometry emerged from the creation of
bubble with black hole inside. The bubble then expands very rapidly until it
reaches the cosmological horizon. This changes the values of the effective CC
from negative to positive values due to the thermalon mediated AdS to dS phase
transition without inclusion of any matter fields coupled to the gravity. This
is the salient feature of this type of gravitational phase transition.

On the other hand, the existence of the graviton mass provides an interesting
results in either quantum theory of gravity and cosmology (see [21, 22] for
reviews). A consideration of the graviton mass has been received attentions
for several decades [23]. In particular, a so-called dRGT massive gravity [24]
becomes a well defined theory of massive gravity. Several problems of the
graviton mass is solved by using the dRGT massive gravity approach, for
instances, redundancy of the spin d.o.f. [25], vDVZ discontinuous [26, 27],
Vainshtein screening [28] and the unstable ghost modes of the massive gravity
[29]. In addition, massive gravity and its extentions can be used to describe
dark energy [30, 31, 32]. Moreover, the dRGT massive gravity is also
extensively studied in various aspects, for examples, black hole solution and
its thermodynamics properties [33], quasinormal modes [34], dark matter [35],
gravitational lensing [36] and wormholes [37].

In this work, we will utilize the thermalon formalism with the higher order
gravity to study the AdS to dS phase transition by including the effect of the
massive gravity in five dimensional spacetime. The main motivation of this
work is that the Gauss-Bonnet term is non-trivial in the spacetime dimension
higher than four [38] and it has been widely used in brane cosmology [39, 40,
41, 42, 43, 44]. Interestingly, one can consider the effects of the graviton
mass since the massive graviton is well defined in the large extra dimension
of the brane cosmology paradigm [45, 46]. Therefore, it is worth exploring the
gravitational phase transition with the massive gravity in five dimensional
spacetime. Charged black holes in Gauss-Bonnet-massive gravity and their
thermodynamic properties is explored in [47]. Phase transition of charged
black hole in Gauss-Bonnet-massive gravity coupled with Yang-Mills field is
investigated in [48]. Moreover, the Gauss-Bonnet gravity in five dimension is
employed to study the thermalon mediated AdS to dS phase transition in various
aspects, e.g., study the critical point of the phase transition in the vacuum
solution [49], phase transition in black hole chemistry picture [50] and
inclusions of Maxwell [51] and Kalb-Ramond [52] fields.

We will organize the paper as follows. The gravitational action of the higher
order gravity with the massive gravity is set up and the corresponding branch
solutions are given in section II. The junction condition between two regions
of AdS and dS vacua is derived and the dynamics of bubble nucleation is
analyzed in section III. In section IV, the gravitational phase transition in
the thermodynamics phase space is studied in details. Finally, we close the
paper with the conclusion in the section V.

##  II The Einstein-Gauss-Bonnet with the presence of the Massive gravity

In this section, we will set up the system of the AdS to dS gravitational
phase transition via the thermalon mediation. The five dimensional action of
Einstein-Gauss-Bonnet-massive gravity (EGBMG) with a presence of the bare CC
($\Lambda$) is given by

$\displaystyle S$ $\displaystyle=\int
d^{5}x\sqrt{-g}\left[R-2\Lambda+\alpha\mathcal{L}_{GB}+m^{2}\mathcal{U}(g,\phi^{a})\right],$
(1)

where $R$ is Ricci scalar, $\alpha$ is the Gauss-Bonnet coupling constant
carrying the mass dimension $-2$ and $m$ can be interpreted as graviton mass.
We have set $16\pi G_{5}=1$ with $G_{5}$ is Newton constant in five dimensions
through out this work. The Gauss-Bonnet term takes a form

$\displaystyle\mathcal{L}_{GB}$
$\displaystyle=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.$
(2)

The graviton potential $\mathcal{U}$ is a function of metric tensor $(g)$ and
Stückelberg field $(\phi^{a})$. It is defined by

$\displaystyle\mathcal{U}$ $\displaystyle=\sum_{i=1}^{4}c_{i}\mathcal{U}_{i},$
(3)

where $c_{i}$ are constants and

$\displaystyle\mathcal{U}_{1}$ $\displaystyle=\left[\mathcal{K}\right],$
$\displaystyle\mathcal{U}_{2}$
$\displaystyle=\left[\mathcal{K}\right]^{2}-\left[\mathcal{K}^{2}\right],$
$\displaystyle\mathcal{U}_{3}$
$\displaystyle=\left[\mathcal{K}\right]^{3}-3\left[\mathcal{K}\right]\left[\mathcal{K}^{2}\right]+2\left[\mathcal{K}^{3}\right],$
$\displaystyle\mathcal{U}_{4}$
$\displaystyle=\left[\mathcal{K}\right]^{4}-6\left[\mathcal{K}^{2}\right]\left[\mathcal{K}\right]^{2}+8\left[\mathcal{K}^{3}\right]\left[\mathcal{K}\right]+3\left[\mathcal{K}^{2}\right]^{2}-6\left[\mathcal{K}^{4}\right].$
(4)

The matrix
$K^{\mu}_{\nu}=\delta^{\mu}_{\nu}-\sqrt{g^{\mu\sigma}f_{ab}\partial_{\sigma}\phi^{a}\partial_{\nu}\phi^{b}}$,
$[\mathcal{K}]=\mathcal{K}^{\mu}_{\mu}$ and
$[\mathcal{K}^{n}]=(\mathcal{K}^{n})^{\mu}_{\mu}$, where
$f_{ab}=diag(0,0,\frac{c^{2}}{1-k\rho^{2}},c^{2}\rho^{2},c^{2}\rho^{2}\sin^{2}\theta)$
is called fiducial metric with constants $c$ and $k$. Varying the action (1)
with respect to $g^{\mu\nu}$ yields Einstein field equations,

$\displaystyle G_{\mu\nu}+\Lambda g_{\mu\nu}+\alpha
T_{\mu\nu}^{GB}+m^{2}T_{\mu\nu}^{MG}$ $\displaystyle=0\,.$ (5)

The effective energy-momentum tensors, $T_{\mu\nu}^{GB}$ and $T_{\mu\nu}^{MG}$
are defined by

$\displaystyle T_{\mu\nu}^{GB}$
$\displaystyle=2\left[R_{\mu\nu}R-\nabla_{\nu}\nabla_{\mu}R\right]+4\left[\nabla^{\rho}\nabla^{\sigma}R_{\mu\rho\nu\sigma}+\nabla^{\sigma}\nabla^{\rho}R_{\mu\rho\nu\sigma}+R_{\mu\rho\sigma\delta}R_{\nu}^{\rho\sigma\delta}\right]$
$\displaystyle~{}~{}~{}-\frac{1}{2}g_{\mu\nu}\left[R^{2}+2R_{\rho\sigma\delta\lambda}R^{\rho\sigma\delta\lambda}-4\Box
R\right],$ $\displaystyle T_{\mu\nu}^{MG}$
$\displaystyle=-\frac{c_{1}}{2}g_{\mu\nu}\left(\mathcal{U}_{1}-\mathcal{K}_{\mu\nu}\right)-\frac{c_{2}}{2}\left(g_{\mu\nu}\mathcal{U}_{2}-2\mathcal{U}_{1}\mathcal{K}_{\mu\nu}+2g_{\rho\nu}\mathcal{K}_{\sigma}^{\rho}\mathcal{K}_{\mu}^{\sigma}\right)$
$\displaystyle-\frac{c_{3}}{2}\left(g_{\mu\nu}\mathcal{U}_{3}+3\mathcal{U}_{2}g_{\mu\nu}+6g_{\rho\nu}\mathcal{U}_{1}\mathcal{K}_{\sigma}^{\rho}\mathcal{K}_{\mu}^{\sigma}-6g_{\rho\nu}\mathcal{K}_{\sigma}^{\rho}\mathcal{K}_{\lambda}^{\sigma}\mathcal{K}_{\mu}^{\lambda}\right)$
$\displaystyle-\frac{c_{4}}{2}\left(g_{\mu\nu}\mathcal{U}_{4}-4\mathcal{U}_{3}\mathcal{K}_{\mu\nu}+12g_{\rho\nu}\mathcal{U}_{2}\mathcal{K}_{\sigma}^{\rho}\mathcal{K}_{\mu}^{\sigma}-24g_{\rho\nu}\mathcal{U}_{1}\mathcal{K}_{\sigma}^{\rho}\mathcal{K}_{\mu}^{\sigma}+24g_{\rho\nu}\mathcal{K}_{\sigma}^{\rho}\mathcal{K}_{\lambda}^{\sigma}\mathcal{K}_{\alpha}^{\lambda}\mathcal{K}_{\mu}^{\alpha}\right).$
(6)

The static spherically symmetric with constant of spatial curvature $k$ is
given by

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+f^{-1}dr^{2}+r^{2}\left(\frac{d\rho^{2}}{1-k\rho^{2}}+\rho^{2}d\theta^{2}+\rho^{2}\sin^{2}\theta
d\phi^{2}\right).$ (7)

The solutions of the metric tensor, $f(r)$ in Eq.(7) is obtained by solving
Einstein field equations (5). It reads

$\displaystyle f_{\pm}$
$\displaystyle=k+\frac{r^{2}}{4\alpha}\left(1\pm\sqrt{1+\frac{16k^{2}\alpha^{2}}{r^{4}}+\alpha\left(\frac{4\Lambda}{3}+\frac{16m_{0}}{r^{4}}\right)-m^{2}\alpha\left(\frac{8cc_{1}}{3r}+\frac{8c^{2}c_{2}}{r^{2}}+\frac{16c^{3}c_{3}}{r^{3}}\right)}\right).$
(8)

Here $m_{0}$ is integration constant which can be interpreted as black hole’s
mass. If we set graviton mass to zero, $m=0$, this solution can be compared
with those found in [49, 50, 51, 52]. Thus, it is more convenient to define
the following quantities

$\displaystyle\alpha\to\frac{\lambda
L^{2}}{2},~{}~{}~{}\Lambda\to\frac{6}{L^{2}},~{}~{}~{}m_{0}\to\frac{\mathcal{M}-\lambda
k^{2}L^{2}}{2},$ (9)

where $L$ is the AdS or dS length scale and $\lambda$ is the dimensionless
Gauss-Bonnet coupling. We also define shorted-hand notation for the massive
graviton term as

$\displaystyle
cc_{1}m^{2}\to\gamma_{a},~{}~{}~{}c^{2}c_{2}m^{2}\to\gamma_{b},~{}~{}~{}c^{3}c_{3}m^{2}\to\gamma_{c}.$
(10)

Therefore, the metric describing AdS outer ($+$) and dS inner ($-$) spacetime
are written by

$\displaystyle f_{\pm}$ $\displaystyle=k+\frac{r^{2}}{2\lambda
L^{2}}\left(1\pm\sqrt{1+4\lambda\left(1+L^{2}\left[\frac{\mathcal{M}}{r^{4}}-\frac{\gamma_{a}}{3r}-\frac{\gamma_{b}}{r^{2}}-\frac{2\gamma_{c}}{r^{3}}\right]\right)}\right).$
(11)

Alternatively, the solution of EGBMG can be obtained by introducing the
following polynomials function

$\displaystyle\Upsilon[g]$ $\displaystyle=-\frac{1}{L^{2}}+g+\lambda
L^{2}g^{2}=\frac{\mathcal{M}}{r^{4}}-\frac{\gamma_{a}}{3r}-\frac{\gamma_{b}}{r^{2}}-\frac{2\gamma_{c}}{r^{3}}.$
(12)

Thus the function $g$ can be explicitly written as

$\displaystyle g_{\pm}$
$\displaystyle=-\frac{1}{2L^{2}\lambda}\left[1\pm\sqrt{1+4\lambda\left(1+L^{2}\left(\frac{\mathcal{M}}{r^{4}}-\frac{\gamma_{a}}{3r}-\frac{\gamma_{b}}{r^{2}}-\frac{2\gamma_{c}}{r^{3}}\right)\right)}\right],$
(13)

or equivalently,

$\displaystyle g_{\pm}$ $\displaystyle=\frac{k-f_{\pm}}{r^{2}}.$ (14)

We will use two branches of solution Eq.(11) of the EGBMG theory for studying
the gravitational phase transition in the next section.

##  III Junction condition and bubble dynamics

In this section, we will define the junction condition connecting two region
(AdS and dS as outside and inside, respectively) of the spacetimes to study
the bubble dynamics. To ensure the continuity between the two spacetimes, the
so-called junction conditions must be satisfied. The junction condition of
EGBMG is defined in terms of co-moving time component of the canonical momenta
($\Pi$) (see detail derivation in Refs.[51, 49, 20])

$\displaystyle\Pi_{\pm}$ $\displaystyle=\frac{2}{3}\lambda
L^{2}g_{\pm}\sqrt{H-g_{\pm}}+\frac{4}{3}H\lambda
L^{2}\sqrt{H-g_{\pm}}+\sqrt{H-g_{\pm}},$ (15)

where $H=\frac{k+\dot{a}^{2}}{a^{2}}$. Here $a\equiv a(\tau)$ is a scaled
factor defined on the induced line elements of the timelike hypersurface
($\Sigma$). We remark that, at the boundaries outer AdS and inner dS
spacetimes, the metric of hypersurface $\Sigma$ takes the similar form (see
more details in [51]). The derivative with respect to co-moving time
coordinate $\tau$ will be denoted by $\dot{a}=\frac{da}{d\tau}$.

The junction condition can be obtained from

$\displaystyle\Pi_{+}^{2}-\Pi_{-}^{2}\rvert_{r_{\pm}=a}$ $\displaystyle=0,$
(16)

where this condition is evaluated at the boundaries between two manifolds
(outer and inner spacetimes). By substituting Eq.(15) into the above
condition, we obtain

$\displaystyle-\dot{a}^{2}$
$\displaystyle=k+\frac{a^{6}}{4L^{2}\lambda\mathcal{A}}\left(g_{+}\left(3+2g_{+}L^{2}\lambda\right)^{2}-g_{-}\left(3+2g_{-}L^{2}\lambda\right)^{2}\right),$
(17)

where

$\displaystyle\mathcal{A}$ $\displaystyle\equiv
3\left(\mathcal{M}_{+}-\mathcal{M}_{-}\right)-a^{3}\left(\gamma_{a+}-\gamma_{a-}\right)-3a^{2}\left(\gamma_{b+}-\gamma_{b-}\right)-6a\left(\gamma_{c+}-\gamma_{c-}\right).$
(18)

The junction condition can be expressed in term of kinetic and effective
potential as

$\displaystyle\frac{1}{2}\dot{a}^{2}+V(a)=0.$ (19)

In order to obtain above equation, we have used the imaginary time, i.e.,
$\tau\to i\tau_{E}$ in the derivative leading to
$(da/d\tau)^{2}\to-(da/d\tau_{E})^{2}$. The effective potential can be shown
explicitly

$\displaystyle V$
$\displaystyle=\frac{a^{6}}{8L^{2}\lambda\mathcal{A}}\left[4\left(2+gL^{2}\lambda\right)\left(\frac{\mathcal{M}}{a^{4}}-\frac{\gamma_{a}}{3a}-\frac{\gamma_{b}}{a^{2}}-\frac{2\gamma_{c}}{a^{3}}\right)+g(1+4\lambda)\right]_{-}^{+}+\frac{k}{2},$
(20)

where we have used the following notation

$\displaystyle\left[\mathcal{O}\right]_{-}^{+}$
$\displaystyle\equiv\mathcal{O}_{+}-\mathcal{O}_{-}.$ (21)

The derivative of effective potential with respect to $a$ is determined by

$\displaystyle V^{\prime}(a)$
$\displaystyle=\frac{a}{8L^{2}\lambda\mathcal{A}^{2}}\Big{(}3a^{5}\Big{[}a^{2}\gamma_{a}+2a\gamma_{b}+2\gamma_{c+}\Big{]}_{-}^{+}\Big{[}g(1+4\lambda)+4(2+gL^{2}\lambda)\Upsilon[g(a)]\Big{]}_{-}^{+}$
$\displaystyle\qquad\qquad\quad+\Big{[}12\mathcal{M}+6a^{4}(1+4\lambda)g-\gamma_{a}a^{3}(13+6gL^{2}\lambda)$
$\displaystyle\qquad\qquad\qquad\quad-6\gamma_{b}a^{2}(5+2gL^{2}\lambda)-6\gamma_{c}a(7+2gL^{2}\lambda)\Big{]}_{-}^{+}\Big{)},$
(22)

The effective potential in Eq.(20) and its derivative in Eq.(22) are useful to
explain the dynamics of the bubble nucleation and the discussion will be given
later. More importantly, we shall assume the continuity of gravity mass by
setting

$\displaystyle\gamma_{a}=\gamma_{a_{+}}=\gamma_{a_{-}},\quad\gamma_{b}=\gamma_{b_{+}}=\gamma_{b_{-}},\quad\gamma_{c}=\gamma_{c_{+}}=\gamma_{c_{-}}.$
(23)

These continuities greatly simplify the expression of effective potential.
Imposing $V(a_{\ast})=V^{\prime}(a_{\ast})=0$, one can explicitly determined
$\mathcal{M}_{\pm}$ at the thermalon location, $a_{\ast}$ as

$\displaystyle\mathcal{M}^{\ast}_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{12\lambda
L^{2}a_{\ast}^{4}}\Big{[}3a_{\ast}^{6}(1+4\lambda)\big{(}6k\lambda
L^{2}+a_{\ast}^{2}(3+2g^{\ast}_{\mp}\lambda L^{2})\big{)}$ (24)
$\displaystyle-2\lambda L^{2}a_{\ast}^{5}\big{(}9k\lambda
L^{2}+a_{\ast}^{2}(4+3g^{\ast}_{\mp}\lambda L^{2})\big{)}\gamma_{a}$
$\displaystyle-12\lambda L^{2}a_{\ast}^{4}\big{(}3k\lambda
L^{2}+a_{\ast}^{2}(1+g^{\ast}_{\mp}\lambda
L^{2})\big{)}\gamma_{b}-12\lambda^{2}L^{4}a_{\ast}^{3}(3k+a_{\ast}^{2}g^{\ast}_{\mp})\gamma_{c}\Big{]},$

where $g^{\ast}_{\pm}=g_{\pm}(a_{\ast})$. From definition of the $\Upsilon$ in
Eq.(12), we obtain

$\displaystyle\Upsilon[g^{\ast}_{\pm}]$
$\displaystyle=C_{1}g^{\ast}_{\pm}+C_{2},$ (25)

where the coefficients $C_{1,2}$ are then determined as

$\displaystyle C_{1}$
$\displaystyle=\frac{1}{2}+\left[2-\frac{L^{2}}{a_{\ast}}\left(\frac{\gamma_{a}}{2}+\frac{\gamma_{b}}{a_{\ast}}+\frac{\gamma_{c}}{a_{\ast}^{2}}\right)\right]\lambda,$
(26) $\displaystyle C_{2}$
$\displaystyle=\frac{3\left(1+4\lambda\right)}{4\lambda
L^{2}}-\frac{3k\left(1+4\lambda\right)}{2a_{\ast}^{2}}+\frac{\gamma_{a}}{a_{\ast}}+\frac{2\gamma_{b}}{a_{\ast}^{2}}+\frac{2\gamma_{c}}{a_{\ast}^{3}}.$
(27)

Thus we can solve these equations simultaneously which yield

$\displaystyle g^{\ast}_{\pm}$
$\displaystyle=-\frac{\left(1+C_{1}\right)\mp\sqrt{1-2C_{1}-3C_{1}^{2}+4\lambda\left(1+C_{2}L^{2}\right)}}{2\lambda
L^{2}}.$ (28)

We note that the $g_{\pm}^{\ast}$ correspond to AdS and dS geometries and the
condition $g_{+}^{\ast}\neq g_{-}^{\ast}$ is required since we study the phase
transition between those spacetimes. Therefore by using Eq.(26–27), the
$g^{\ast}_{\pm}$ can be expressed in terms of
$k,\lambda,L^{2},\gamma_{a},\gamma_{b}$ and $\gamma_{c}$. After deriving all
relevant equations, we are ready to analyze the bubble dynamics.

Figure 1: Effective potential plotted as function of $a$ for $a_{\ast}=1$,
$L=1$, $k=1$: Left fixed $\lambda=0.015$ and $\gamma_{i}=$ 0, 2.5, 5, 7.5, 10
(red, green, blue, black, magenta), Right fixed $\gamma_{i}=5$ and $\lambda=$
0.001, 0.003, 0.005, 0.007, 0.01 (red, green, blue, black, magenta), where
$i=$ {$a$, $b$, $c$}

From now on, the spherical bubble, $k=1$ will be studied throughout this work.
In Fig. 1, we display an example of behaviour of effective potential against
$a$. In this figure, we fix $\gamma_{a}=\gamma_{b}=\gamma_{c}$. We vary
$\gamma_{i}$ while keep other parameters fixed in the left figure. The right
figure shows the effective potential with $\lambda$ is varied. Despite having
various parameters, throughout this work, we shall particularly focus on the
set of parameter which yield $\mathcal{M}^{*}_{\pm}$ are real positive value.
As the results, the Fig. 1 clearly shows that the effective potential has the
maximum value at the given thermalon position, $a_{\ast}=1$. This means that
the thermalon is in the unstable mode and ready to roll down from the top of
the effective potential. Consequently, the bubble, representing dS space with
true vacuum can expand and leading to the AdS to dS phase transition
eventually. In Fig. 2, parameter spaces where $\mathcal{M}^{*}_{\pm}>0$ are
illustrated. In each plots, we fix bubble location and AdS (dS) length scale
to be unity. The GB coupling constant is chosen to be 0.1. With given
$\gamma_{c}$, allowed value of $\gamma_{a},\gamma_{b}$ can be obtained. The
shaded regions increase as $\gamma_{c}$. We should emphasize that the plots in
Fig. 2 are only partially displayed for demonstration purpose. From our
numerical investigation, the shaded regions can be extended rightward and
upward.

Figure 2: Possible parameter space between $\gamma_{a}-\gamma_{b}$ for fixed
$a_{\ast}=1$, $L=1$, $k=1$, $\lambda=0.1$: Left $\gamma_{c}=-1$, Middle
$\gamma_{c}=0$, Right $\gamma_{c}=1$.

In general, the bubble can be either expanded or collapsed after rolling down
from the effective potential. However, we are interested only the expanding
bubble case which can change the values of the effective CC from the false
vacuum AdS space to the true vacuum dS space asymptotically. Therefore, a
consideration of the collapsing bubble case is neglected in this study. One
can see Ref.[20] for more detail discussions of the collapsing bubble that
leads to violation of the cosmic censorship hypothesis.

We close this section by providing a brief discussion on how we ensure that
the bubble will expand to infinity reaching the cosmic horizon and changing
the effective CC to the new value a whole spacetime. By considering the
definitions of the canonical momenta of the AdS (outside) and dS (inside)
spaces and the junction condition in (15) and (16), respectively, we find,

$\displaystyle
H\approx\frac{1}{2\left(\mathcal{M}_{+}-\mathcal{M}_{-}\right)}\left[a^{4}\int_{g_{-}}^{g_{+}}dg\,\Upsilon\big{[}g\big{]}-\Big{[}g\left(\mathcal{M}-a^{3}\gamma_{a}-a^{2}\gamma_{b}-a\gamma_{c}\right)\Big{]}_{-}^{+}\right].$
(29)

Here the above expression is obtained up to the first order expansion of the
$1/H$. We can see explicitly that $H\equiv\dot{a}/a$ (speed of the bubble
expansion) approaches to $\infty$ when $a\to\infty$. This corresponds to the
the bubble expansion to infinity at the finite time.

##  IV Gravitational phase transition

This section presents a central result of this work on the AdS to dS phase
transition mediated by thermalon in EGBMG theory. This section is divided into
two parts. First, we discuss how we obtain physically corrected thermalon
configuration. Second, from a set of parameter in EGBMG which allowed
thermalon configuration, we analyse thermodynamic phase space of the
gravitational phase transition.

###  IV.1 Thermalon configurations, horizons and Nariai bound

Since we need to ensure that the thermalon’s location $a_{\ast}$ lies in the
casual connected region outside the black hole. Thus, it is convenient to
firstly determine the casual structure of the metric Eq.(8). The horizon’s
location can be obtained by imposing $f_{-}(r_{H})=0$. This yields quartic
equation

$\displaystyle
r_{H}^{4}-\frac{\gamma_{a}L^{2}}{3}r_{H}^{3}-\left(1+\gamma_{b}\right)L^{2}r_{H}^{2}-2\gamma_{c}L^{2}r_{H}+L^{2}\left(M^{\ast}_{-}-\lambda
L^{2}\right)$ $\displaystyle=0.$ (30)

The roots of this equation can be interpreted as the dS branch (inner
spacetime) horizon. In general, there will be four possible roots. However,
only real positive roots are physically relevant. An example plot of metric
function $f$ is illustrated in fig. 3. In these plots, the bubble position is
always fixed to unity i.e., $a_{\ast}=1$. Two roots of $f_{-}$ correspond to
event horizon ($r_{H}$) and cosmological horizon ($r_{C}$). We observe that as
$\gamma_{a},\gamma_{b}$ increases the metric function also increases. In
addition, increasing in $\gamma_{a},\gamma_{b}$ does not significantly alter
the location of the event horizon while the cosmological horizon is shifted to
a larger value. Moreover as we expect, the radius of cosmological horizon
becomes larger as $L$ is increased.

Figure 3: Plot of $f_{-}$ when $a_{\ast}=1,k=1$ Red:
$L=1,\lambda=0.2,\gamma_{c}=1$ (dashed) $\gamma_{a}=0,\gamma_{b}=0$ (solid)
$\gamma_{a}=0.6,\gamma_{b}=0.5$ and Blue: $L=3,\lambda=0.015,\gamma_{c}=0.5$
(dashed) $\gamma_{a}=0,\gamma_{b}=0$ (solid) $\gamma_{a}=1.6,\gamma_{b}=0.8$.
The subplot displays a close up behaviour of metric function $f$.

Equation (30) is in a standard form of quartic equation
$ar^{4}+br^{3}+cr^{2}+dr+e=0.$ The roots of this equation are given by

$\displaystyle r_{1,2}$
$\displaystyle=-\frac{b}{4a}-S\pm\frac{1}{2}\sqrt{-4S^{2}-2p+\frac{q}{S}},$
(31) $\displaystyle r_{3,4}$
$\displaystyle=-\frac{b}{4a}+S\pm\frac{1}{2}\sqrt{-4S^{2}-2p-\frac{q}{S}},$
(32)

where we have defined the following parameters

$\displaystyle p$
$\displaystyle\equiv\frac{8ac-3b^{2}}{8a^{2}}=-\frac{L^{2}}{24}\left(L^{2}\gamma_{a}^{2}+24\left(1+\gamma_{b}\right)\right),$
$\displaystyle q$
$\displaystyle\equiv\frac{b^{3}-4abc+8a^{2}d}{8a^{3}}=-\frac{L^{2}}{216}\left(L^{4}\gamma_{a}^{3}+36L^{2}\gamma_{a}\left(1+\gamma_{b}\right)+432\gamma_{c}\right),$
$\displaystyle S$
$\displaystyle\equiv\frac{1}{2}\sqrt{\frac{1}{3a}\left(P+\frac{\delta_{0}}{P}\right)-\frac{2p}{3}},$
$\displaystyle P$
$\displaystyle\equiv\left(\frac{\delta_{1}+\sqrt{\delta_{1}^{2}-4\delta_{0}^{3}}}{2}\right)^{1/3},$
$\displaystyle\delta_{0}$ $\displaystyle\equiv
c^{2}-3bd+12ae=12\mathcal{M}^{*}_{-}L^{2}+\left(\left(1+\gamma_{b}\right)^{2}-2\gamma_{a}\gamma_{c}-12\lambda\right)L^{4},$
$\displaystyle\delta_{1}$ $\displaystyle\equiv
2c^{3}-9bcd+27b^{2}e+27ad^{2}-72ace=\left(72\mathcal{M}^{*}_{-}\left(1+\gamma_{b}\right)+108\gamma_{c}^{2}\right)L^{4}$
$\displaystyle+\left(3\mathcal{M}^{*}_{-}\gamma_{a}^{2}-2\left(1+\gamma_{b}\right)\left(1+\gamma_{b}\left(2+\gamma_{b}\right)-3\gamma_{a}\gamma_{c}+36\lambda\right)\right)L^{6}-3\lambda\gamma_{a}^{2}L^{8}.$
(33)

Thus we can now plot the horizon radius as a function of thermalon’s location
$a_{\ast}$ and this is demonstrated in fig. 4. To be physically relevant, we
pay our attention only to the case where the thermalon is located within the
casual region of inner spacetime i.e. $r_{H}\leq a_{\ast}\leq r_{C}$. In
practice, we fix the maximum value of bubble location at $a_{\ast}=1$ where
the spacetime approaches the Narai limit $r_{H}\sim r_{C}$. We remark that,
throughout this work, we shall especially focus on the case where 1) both
$\mathcal{M}^{*}_{\pm}$ are real positive. 2) the metric function Eq.(8) has
at least two positive real roots. 3) The thermalon always stay between black
hole’s event horizon and cosmological horizon. We need to ensure that these
three conditions are met throughout our investigation.

Figure 4: Location of horizons as a function of $a_{\ast}$ when $k=1$, $L=1$,
$\lambda=0.2$, Left: $\gamma_{a}=-2.393,\gamma_{b}=2.197,\gamma_{c}=0$,
Middle: $\gamma_{a}=-0.931$, $\gamma_{b}=-0.035$, $\gamma_{c}=1.5$, Right:
$\gamma_{a}=0.532$, $\gamma_{b}=-2.266$, $\gamma_{c}=3$. The red, black and
blue curved are event horizon, thermalon (bubble) location and cosmological
horizon respectively.

###  IV.2 Thermodynamics of the AdS to dS phase transition mediated by
thermalon

This subsection aims to investigate the gravitational AdS to dS phase
transition in the thermodynamics phase space. We follow Refs.[49, 20, 19, 50,
52, 51] for all formalisms in this study. The gravitational phase transition
in this work is a generalization of the well known Hawking-Page (HP) phase
transition [53] where at the critical temperature the thermal AdS space
becomes unstable and then decays to the the AdS black hole with the lower free
energy. We employ the generalized HP phase transition in this work. The decay
mechanism proceeds via nucleation of the bubbles or the thermalon mediation
where the initial false vacuum state is the thermal AdS. When it is unstable
and decay into black hole inside the bubble of true vacuum dS space at the
critical temperature. This phase transition is produced by the quantum
tunneling or jumping over the wall of the quasi particle state in the
Euclidean sector of the bubbles called the thermalon [18, 19, 20]. In other
words, the thermalon effectively changes the branch of the false vacuum AdS
solution to another true vacuum dS solution.

Taking this phase transition mechanism into account, one can calculate the
thermodynamics properties of the thermalon in the same fashion as black hole
thermodynamics are calculated. The Euclidean action of the thermalon,
$\mathcal{I}_{E}$ has been derived in [19, 20] and it is given by

$\displaystyle\mathcal{I}_{E}=\beta_{+}\,\mathcal{M}_{+}+S\,,$ (34)

where $\beta_{+}$, $\mathcal{M}_{+}$ and $S$ are the inverse Hawking
temperature, mass of the external observer in the asymptotic thermal AdS and
the entropy of the dS black hole, respectively. The Gibbs free energy of the
thermalon in the canonical ensemble is written by [49, 20, 54],

$\displaystyle F$ $\displaystyle=\mathcal{M}_{+}-T_{+}S.$ (35)

To obtain the Gibbs free energy of the thermalon, one can use the on-shell
regularization method by subtracting the thermal AdS space background [49, 20,
50]. More importantly, there are two competing phases in this system i.e., the
thermal AdS (outside) and the thermalon hosting the black hole in the dS space
(inside). Then, we will study the phase transition by comparing the Gibbs free
energy of the thermalon to the thermal AdS one. In practice, we can set the
Gibbs free energy of the thermal AdS space as zero ($F_{\rm AdS}=0$) since
$F_{\rm AdS}$ has already considered as the background subtraction. The
Hawking temperature of inner five dimensional dS black hole in EGBMG can be
defined by [55]

$\displaystyle T_{-}$ $\displaystyle=\frac{1}{4\pi
r_{H}}\left[\sum_{n=0}^{2}\frac{kc_{n}\left(4-2n\right)}{\left(kr_{H}^{-2}\right)^{1-n}}+\gamma_{a}r_{H}+2\gamma_{b}+\frac{2\gamma_{c}}{r_{H}}\right]\left[\sum_{n=0}^{2}\frac{nc_{n}}{\left(kr_{H}^{-2}\right)^{1-n}}\right]^{-1},$
(36)

where

$\displaystyle c_{0}$
$\displaystyle=-L^{-2},~{}~{}~{}~{}~{}c_{1}=1,~{}~{}~{}~{}~{}c_{2}=\lambda
L^{2}.$

Moreover, the Bekenstein-Hawking entropy is given by [55]

$\displaystyle S$ $\displaystyle=4\pi
r_{H}^{5}\sum_{n=0}^{2}\frac{nc_{n}\left(kr_{H}^{-2}\right)^{n}}{k\left(5-2n\right)}.$
(37)

The Hawking temperature of outer AdS spacetime is computed by

$\displaystyle T_{+}$
$\displaystyle=\frac{\sqrt{f_{+}(a_{\ast})}}{\sqrt{f_{-}(a_{\ast})}}T_{-}.$
(38)

Figure 5: The Hawking temperature (left) and free energy (right) of thermalon
configuration as function of $a$ for various value of $\lambda$ with $k=1$,
$L=1$, $\gamma_{a}=-2.296$, $\gamma_{b}=2.048$ and $\gamma_{c}=0.1$. Figure 6:
Free energy $F$ of the thermalon configuration as a function of the
themperature $T_{+}$ for various value of $\lambda$ with $k=1$, $L=1$,
$\gamma_{a}=-2.296$, $\gamma_{b}=2.048$ and $\gamma_{c}=0.1$. The subplot
displays the behaviour of free energy at larger value of $\lambda$. The
colored code can be found in Fig. 5

We plot $T_{+}$ and $F$ as a function of the thermalon radius, $a$ and the
results are shown in Fig. 5. In Fig. 5, the temperature and free energy are
plotted as a function of the thermalon’s location. It is clear that the
temperature increases with bubble position. As $\lambda$ increases, the slope
of $T_{+}$ becomes more flatten. Moreover, we observe phase transition in free
energy plots. As $a$ increases, the thermalon configuration becomes more
thermodynamically preferred. The phase transition is apparently more obvious
as $\lambda$ is larger. The phase transition occurs at larger bubble’s
location as $\lambda$ increases. Additionally, we observe that the physically
relevant thermalon configuration exists at a smaller thermalon’s location $a$
at lower $\lambda$.

By substituting results in Eqs.(24,36,38,37) in the thermalon free energy
Eq.(35), we are ready to study the thermodynamics phase space between $F$ as
function of $T_{+}$. The relation between the Gibbs free energy and Hawking
temperature $T_{+}$ is illustrated in Fig. 6. In this figure, the parameter
$\lambda$ is varied while the others are kept fixed. As $\lambda$ increases,
the free energy become less dependent on temperature (the slope becomes more
inclined). According to our numerical investigation, we particularly choose
the value of $\gamma_{a}=-2.296,\gamma_{b}=2.048$ and $\gamma_{c}=0.1$ such
that the thermalon’s location lies between the cosmological and event
horizons. At small $\lambda$, the minimum value of $F$ is increasing as
$\lambda$ becomes larger. However, the minimum starts to decrease from
$\lambda=0.25$ until it reaches the lowest value at $\lambda=1.2$. After that,
the minimum of free energy rapidly increases as $\lambda$ increases.

As a result, we find that thermalon always produces the AdS to dS phase
transition since all Gibbs free energies of the thermalon are negative at
their maximum temperature for all possible values of the given Gauss-Bonnet
coupling, $\lambda$. This is an interesting result in this work because
usually, the Gauss-Bonnet coupling, $\lambda$ plays the role as the order
parameter of the AdS to dS phase transition in both without [49, 50] and with
the presences of the gauge fields [51, 52]. Our results suggest that with a
proper value of $\gamma_{a,b,c}$, the gravitational phase transition from AdS
to dS is inevitable. This means that in this scenario, the couplings of the
massive gravity, ($\gamma_{a,b,c}$) play the role as the order parameter
rather than the Gauss-Bonnet coupling, ($\lambda$) in this type of phase
transition.

##  V Conclusion

In recent years, the most widely studied theory in the higher curvature
gravities is known as Einstein-Gauss-Bonnet (EGB) theory of gravity. The EGB
theory of gravity is a special case of the Lovelocks gravity. The GB term is
only nontrivial in spacetime dimensions $(4+1)D$ or greater. In this work, we
have studied gravitational phase transition from AdS to dS geometries by
engaging the well-defined massive theory of gravity to the EGB one. This
inclusion basically features a matter field sector in the theory.

The thermalon induces the decay of the negative effective cosmological
constant to the positive one due to the higher-order gravity effects and the
decay mechanism proceeds through nucleation of the bubbles or the thermalon of
true vacuum (dS) inside the false vacuum (thermal AdS) Refs.[19, 51]. It was
demonstrated that the probability of the decay, $P$, of the thermalon
effectively jumping from AdS to dS branch solutions is governed by $P\propto
e^{I_{E}}$ with $I_{E}$ being the Euclidean action difference between initial
thermal AdS and the thermalon (bubble state). As a result, the system will end
up in the stable dS black hole after the thermalon expansion reaching the
asymptotically dS region in a finite time.

We have also determined the thermodynamics phase space of the Hawking
temperature and free energy and found that the free energy of the thermalon is
always negative at the maximum of Hawking temperature for all specific values
of the parameters of the massive gravity sector. Additionally, we have also
discovered that a generic behavior of the phase transition in the EGBMG theory
is affected by the specific values of the parameters in the massive gravity
instead of the GB coupling in contrary to those cases found in the previous
publications in which the authors have studied the vacuum and the charged
cases.

With presence of the massive gravity sector, we have demonstrated that the
phase transition mediated by thermalon from AdS to dS asymptotics is
inevitable in the present work. and specific valued of the parameters in the
massive gravity behave as the order parameters in the phase transition instead
of the Gauss-Bonnet coupling. Interestingly, our results are in line with the
claim that the generalized gravitational phase transition is a generic
behavior of the higher-order gravity theories when the matter field is
included.

###### Acknowledgements.

This work (Grant No. RGNS 64-217) was financially supported by Office of the
Permanent Secretary, Ministry of Higher Education, Science, Research and
Innovation. We are graceful to Chakrit Pongkitivanichkul for useful
discussion. P. Channuie acknowledged the Mid-Career Research Grant 2020 from
National Research Council of Thailand (NRCT5-RSA63019-03). D. Samart is
financially supported by the Mid-Career Research Grant 2021 from National
Research Council of Thailand under a contract No. N41A640145. S.P. would like
to acknowledge Piyabut Burikham for his advice and support.

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# On transformation formulas of $p$-adic hypergeometric functions
$\mathscr{F}_{a,...,a}^{(\sigma)}(t)$ and
$\widehat{\mathscr{F}}_{a,...,a}^{(\sigma)}(t)$

Wang Chung-Hsuan

###### Abstract

In this paper, we recall hypergeometric functions $\mathscr{F}^{\rm
Dw}_{a_{1},\cdots,a_{s}}(t),$
$\mathscr{F}^{(\sigma)}_{a_{1},\cdots,a_{s}}(t)$,
$\widehat{\mathscr{F}}^{(\sigma)}_{a,\cdots,a}(t)$ and the conjecture of their
transformation formulas. Then we prove that one of transformation formula
implies another.

Keywords— $p$-adic hypergeometric functions, $p$-adic hypergeometric functions
of logarithmic type, congruence relations

##  1 Introduction

The classical hypergeometric function of one-variable is defined to be the
power series

${}_{s}F_{s-1}\left({a_{1},\ldots,a_{s}\atop
b_{1},\ldots,b_{s-1}};t\right)=\sum\limits_{k=0}^{\infty}\frac{(a_{1})_{k}\cdots(a_{s})_{k}}{(b_{1})_{k}\cdots(b_{s-1})_{k}}\frac{t^{k}}{k!}$

where $(\alpha)_{k}$ denotes the Pochhammer symbol (cf. [3]). Let
$(a_{1},...,a_{s})\in\mathbb{Z}_{p}^{s}$ be a $s$-tuple of $p$-adic integers,
and consider the series

$F_{a_{1},...,a_{s}}(t)={}_{s}F_{s-1}\left({a_{1},\ldots,a_{s}\atop
1,\ldots,1};t\right)=\sum\limits_{k=0}^{\infty}\frac{(a_{1})_{k}}{k!}\cdots\frac{(a_{s})_{k}}{k!}t^{k}.$

This is a formal power series with $\mathbb{Z}_{p}$-coefficients. In the paper
[2], B. Dwork introduced his $p$-adic hypergeometric functions
$\mathscr{F}^{\rm Dw}_{a_{1},...,a_{s}}(t)$ (Definition 4.3) and showed that
$\mathscr{F}^{\rm Dw}_{a_{1},...,a_{s}}(t)$ satisfy congruence relation ([2,
p.41, Theorem 3])

$\mathscr{F}^{\rm
Dw}_{a_{1},...,a_{s}}(t)\equiv\frac{[F_{a_{1},\cdots,a_{s}}(t)]_{<p^{n}}}{[F_{a_{1}^{\prime},\cdots,a_{s}^{\prime}}(t^{p})]_{<p^{n}}}\mod{p^{n}\mathbb{Z}_{p}[[t]]}$

where for a power series $f(t)=\sum{c_{n}}t^{n}$, we write
$[f(t)]_{<m}:=\sum_{n<m}c_{n}t^{n}$ the truncated polynomial.

In [1] and [4], the $p$-adic hypergeoemtric functions of logarithmic type
$\mathscr{F}^{(\sigma)}_{a_{1},\cdots,a_{s}}(t)$ (Definition 2.1) and
$\widehat{\mathscr{F}}_{a,...,a}^{\;(\sigma)}(t)$ (Definition 2.2) are
introduced. And it is shown that these $p$-adic hypergeometric functions
satisfy congruence relations similar to Dwork’s (Theorem 2.4 and Theorem 2.5).
In [1], Asakura studied his functions from the viewpoint of $p$-adic
regulators. In particular, some values of
$\mathscr{F}_{a_{1},...,a_{s}}^{(\sigma)}(t)$ are expected to be the special
values of $p$-adic $L$-function of elliptic curves over $\mathbb{Q}$ ([1,
§5]).

Furthermore, two conjectures between these hypergeometric functions are
proposed in [4]. It is called the transformation formula of Dwork’s $p$-adic
hypergeometric functions (Conjecture 4.4)

$\mathscr{F}^{{\rm Dw}}_{a,\cdots,a}(t)=((-1)^{s}t)^{l}\mathscr{F}^{{\rm
Dw}}_{a,\cdots,a}(t^{-1})$

and the transformation formula of $\mathscr{F}_{a,...,a}^{(\sigma)}(t)$ and
$\widehat{\mathscr{F}}_{a,...,a}^{\;(\sigma)}(t)$ (Conjecture 4.1)

$\mathscr{F}_{a,\cdots,a}^{\;(\sigma)}(t)=-\widehat{\mathscr{F}}_{a,\cdots,a}^{\;(\widehat{\sigma})}(t^{-1}).$

In [4], some special cases of these transformation formula have been proved
(the case $s=2$, $a\in\frac{1}{N}{\mathbb{Z}}$, $0<a<1$ and $p>N$, [4, Theorem
4.14] and [4, Theorem 4.20]).

The main result (Theorem 4.9) of this paper is that we prove a slight revision
of Conjecture 4.4(Conjecture 4.7) implies Conjecture 4.1. Using this theorem,
we can show as a corollary that Conjecture 4.1 is true when $s=1$.

This paper is organized as follows. In §2, we recall the definition and some
properties of hypergeometric functions $\mathscr{F}_{a,...,a}^{\;(\sigma)}(t)$
and $\widehat{\mathscr{F}}_{a_{1},...,a_{s}}^{\;(\sigma)}(t)$. In §3, we prove
some relation between coefficients of these hypergeometric functions. In §4,
we first recall the conjectures between these hypergeometric functions and
then move on to the transformation formula of Dwork’s $p$-adic hypergeometric
functions. Finally, we give the proof of our main result using the relation in
§3.

Acknowledgement. I would like to thank for the help of Professor Po-Yi Huang.
He gave a lot of helpful advice and comments on this paper.

##  2 Definition of $\mathscr{F}_{a,...,a}^{\;(\sigma)}(t)$ and
$\widehat{\mathscr{F}}_{a_{1},...,a_{s}}^{\;(\sigma)}(t)$

In this section, we recall the definition of $p$-adic hypergeometric functions
$\mathscr{F}_{a_{1},...,a_{s}}^{\;(\sigma)}(t)$ and
$\widehat{\mathscr{F}}_{a,...,a}^{\;(\sigma)}(t)$.

Let $W=W(\overline{\mathbb{F}}_{p})$ denote the Witt ring, and $K$=Frac$W$ its
fractional field. Let $\sigma:W[[t]]\rightarrow W[[t]]$ be a $p$-th Frobenius
given by $\sigma(t)=ct^{p}$ with $c\in 1+pW$ :

$\biggl{(}\sum_{i}a_{i}t^{i}\biggr{)}^{\sigma}=\sum_{i}a_{i}^{F}c^{i}t^{ip}$
(2.1)

where $F:W\rightarrow W$ is the Frobenius on $W$. Given $a\in\mathbb{Z}_{p}$.
Let $a^{\prime}$ be the Dwork prime of $a$, which is defined to be $(a+l)/p$
where $l\in\\{0,1,...,p-1\\}$ is the unique integer such that $a+l\equiv 0\mod
p$. We denote the $i$-th Dwork prime by $a^{(i)}$ which is defined to be
$(a^{(i-1)})^{\prime}$ with $a^{(0)}=a.$

Fix an $s\in\mathbb{Z}_{>0}$. Let

$F_{a_{1},...,a_{s}}(t)=\sum_{k=0}^{\infty}\frac{(a_{1})_{k}}{k!}\cdots\frac{(a_{s})_{k}}{k!}t^{k}$
(2.2)

be the hypergeometric series for $a\in\mathbb{Z}_{p}$, where $(a)_{k}$ denotes
the Pochhammer symbol (i.e. $(\alpha)_{k}=\alpha(\alpha+1)\cdots(\alpha+k+1)$
when $k\neq 0$ and $(\alpha)_{0}=1$).

Put

$q:=\left\\{\begin{aligned} 4&\quad p=2\\\ p&\quad p\geq 3.\\\
\end{aligned}\right.$

and

$e:=l^{\prime}-\lfloor\frac{l^{\prime}}{p}\rfloor.$

where $l^{\prime}\in\\{0,1,...,q-1\\}$ be the unique integer such that
$a+l^{\prime}\equiv 0\mod q$.

######  Definition 2.1 ([1, Definition 3.1], $p$-adic hypergeometric functions
of logarithmic type).

Let $s\geq 1$ be a positive integer. Let
$(a_{1},\cdots,a_{s})\in\mathbb{Z}_{p}$ and
$(a_{1}^{\prime},\cdots,a_{s}^{\prime})$ where $a_{i}^{\prime}$ denotes the
Dwork prime. Let $\sigma:W[[t]]\rightarrow W[[t]]$ be the $p$-th Frobenius
endomorphism given by $\sigma(t)=ct^{p}$ with $c\in 1+pW.$ Then we define the
$p$-adic hypergeometric functions of logarithmic type to be

$\mathscr{F}^{(\sigma)}_{a_{1},\cdots,a_{s}}(t):=\frac{G^{(\sigma)}_{a_{1},\cdots,a_{s}}(t)}{F_{a_{1},\cdots,a_{s}}(t)}$

where

$G^{(\sigma)}_{a_{1},\cdots,a_{s}}(t)=\psi_{p}(a_{1})+\cdots+\psi_{p}(a_{s})+s\gamma_{p}-p^{-1}\log(c)+\int_{0}^{t}(F_{a_{1},\cdots,a_{s}}(t)-F_{a_{1}^{\prime},\cdots,a_{s}^{\prime}}(t^{\sigma}))\frac{dt}{t},$

$\psi_{p}(z)$ is the $p$-adic digamma function defined in [1, §2.2],
$\gamma_{p}$ is the $p$-adic Euler constant ([1, §2.2]) and $\log(z)$ is the
Iwasawa logarithmic function.

######  Definition 2.2 ([4, Definition 2.1]).

Let $a\in\mathbb{Z}_{p}\backslash\mathbb{Z}_{\leq 0}$ and $\sigma:W[[t]]\to
W[[t]]$ be the $p$-th Frobenius endomorphism given by $\sigma(t)=ct^{p}$ with
$c\in 1+pW$. Then we define

$\widehat{\mathscr{F}}_{a,...,a}^{\;(\sigma)}(t):=\frac{\widehat{G}^{(\sigma)}_{a,...,a}(t)}{F_{a,...,a}(t)}$

where

$\widehat{G}^{(\sigma)}_{a,...,a}(t)=t^{-a}\int_{0}^{t}(t^{a}F_{a,...,a}(t)-(-1)^{se}[t^{a^{\prime}}F_{a^{\prime},...,a^{\prime}}(t)]^{\sigma})\frac{dt}{t}.$

######  Remark 2.3.

Here we think $t^{\alpha}$ to be an abstract symbol with relations
$t^{\alpha}\cdot t^{\beta}=t^{\alpha+\beta}$, on which $\sigma$ acts by
$\sigma(t^{\alpha})=c^{\alpha}t^{p\alpha}$, where

$c^{\alpha}=(1+pu)^{\alpha}:=\sum_{i=0}^{\infty}\binom{\alpha}{i}p^{i}u^{i},\quad
u\in W.$

Moreover we think $\int_{0}^{t}(-)\frac{dt}{t}$ to be a operator such that

$\int_{0}^{t}t^{\alpha}\frac{dt}{t}=\frac{t^{\alpha}}{\alpha},\quad\alpha\neq
0.$

In what follows, we fix an $a\in\mathbb{Z}_{p}\backslash\mathbb{Z}_{\leq 0}$
and for simplicity we write $F_{a}(t):=F_{a,\cdots,a}(t)$ (similar for
$G^{(\sigma)}_{a,\cdots,a}(t),$ $\widehat{G}^{(\sigma)}_{a,\cdots,a}(t),$
$\mathscr{F}^{(\sigma)}_{a,\cdots,a}(t)$ and
$\widehat{\mathscr{F}}^{\;(\sigma)}_{a,\cdots,a}(t)).$

If we write $F_{a}(t)=\sum A_{k}t^{k}$, $F_{a^{\prime}}(t)=\sum
A_{k}^{(1)}t^{k}$, $G^{(\sigma)}_{a}(t)=\sum B^{(\sigma)}_{k}t^{k}$ and
$\widehat{G}^{(\sigma)}_{a}(t)=\sum\widehat{B}_{k}^{(\sigma)}t^{k}$, then

$B_{0}^{(\sigma)}=\psi_{p}(a_{1})+\cdots+\psi_{p}(a_{s})+s\gamma_{p}-p^{-1}\log(c),\quad
B_{k}^{(\sigma)}=\frac{A_{k}-c^{k/p}A^{(1)}_{k/p}}{k},\quad
k\in\mathbb{Z}_{>0}$

and

$\widehat{B}_{k}^{(\sigma)}=\frac{1}{k+a}\biggl{(}A_{k}-(-1)^{se}(A_{\frac{k-l}{p}}^{(1)})c^{\frac{k+a}{p}}\biggr{)},$

where $A_{\frac{m}{p}}^{(1)}=0$ if $m\not\equiv 0\mod p$ or $m<0$. In fact, we
have $\mathscr{F}_{a}^{\;(\sigma)}(t)$, $G^{(\sigma)}_{a}(t)$,
$\widehat{\mathscr{F}}_{a}^{\;(\sigma)}(t)$ and
$\widehat{G}^{(\sigma)}_{a}(t)$ are power series with $W$-coefficients.([4,
Lemma 2.2])

Now let us recall so-called congruence relations of
$\mathscr{F}^{(\sigma)}_{a}(t)$ and
$\widehat{\mathscr{F}}^{\;(\sigma)}_{a}(t).$ For a power series
$f(t)=\sum^{\infty}_{i=0}a_{i}t^{i}\in W[[t]]$, we denote
$f(t)_{<n}:=\sum_{i=0}^{n-1}a_{i}t^{i}$ the truncated polynomial.

######  Theorem 2.4 (congruence relation of $\mathscr{F}^{(\sigma)}_{a}(t)$).

Let $a\in\mathbb{Z}_{p}\backslash\mathbb{Z}_{\leq 0}.$ Then

$\mathscr{F}^{(\sigma)}_{a}(t)\equiv\frac{G^{(\sigma)}_{a}(t)_{<p^{n}}}{F_{a}(t)_{<p^{n}}}\mod
p^{n}W[[t]]$

if $c\in 1+qW.$ If $p=2$ and $c\in 1+2W$, then the congruence holds modulo
$p^{n-1}.$

###### Proof.

This is [1, Theorem 3.3]. ∎

######  Theorem 2.5 (congruence relation of
$\widehat{\mathscr{F}}^{\;(\sigma)}_{a}(t)$).

Let $a\in\mathbb{Z}_{p}\backslash\mathbb{Z}_{\leq 0}$ and suppose that $c\in
1+qW.$ Then

$\widehat{\mathscr{F}}^{\;(\sigma)}_{a}(t)\equiv\frac{\widehat{G}^{(\sigma)}_{a}(t)_{<p^{n}}}{F_{a}(t)_{<p^{n}}}\mod
p^{n}W[[t]]$

for all $n\in\mathbb{Z}_{\geq 0}.$

###### Proof.

This is [4, Theorem 3.1]. ∎

##  3 Congruence Relations for $B_{k}/A_{k}$ and $\widehat{B}_{k}/A_{k}$

Here we put $\sigma:W[[t]]\rightarrow W[[t]]$ and
$\widehat{\sigma}:W[[t]\rightarrow W[[t]]$ by $\sigma(t)=ct^{p}$ and
$\widehat{\sigma}(t)=c^{-1}t^{p}$ where $c\in 1+pW.$ For simplicity, in what
follows, we write $B_{k}$ for $B_{k}^{(\sigma)}$ and $\widehat{B}_{k}$ for
$\widehat{B}_{k}^{(\widehat{\sigma})}.$

Let us recall the following properties for $B_{k}/A_{k}$ and
$\widehat{B}_{k}/A_{k}$.

######  Theorem 3.1.

$B_{k}/A_{k}$ and $\widehat{B}_{k}/A_{k}$ are in the Witt ring
$W(\overline{\mathbb{F}}_{p})$. Furthermore, for any
$k,k^{\prime}\in\mathbb{Z}_{\geq 0}$ and $n\in\mathbb{Z}_{>0}$, we have

$\frac{B_{k}}{A_{k}}\equiv\frac{B_{k^{\prime}}}{A_{k}}\mod
p^{n},\quad\frac{\widehat{B_{k}}}{A_{k}}\equiv\frac{\widehat{B}_{k^{\prime}}}{A_{k}}\mod
p^{n}$

if $k\equiv k^{\prime}\mod p^{n}.$

###### Proof.

This is [1, Lemma 3.9] and [4, Lemma 3.5]. Although, in [1, Lemma 3.9], the
author only treated the case $c=1,$ it can be shown that (as in [4, Lemma
3.7]) that it is true for general $c\in 1+pW.$ ∎

Let us define two functions $f:\mathbb{Z}_{>0}\rightarrow W$ and
$\widehat{f}:\mathbb{Z}_{>0}\rightarrow W$ by

$f(k)=\frac{B_{k}}{A_{k}},\quad\widehat{f}(k)=\frac{\widehat{B}_{k}}{A_{k}}.$

Then by Theorem 3.1 and using $p$-adic interpolation, we can extend functions
$f$ and $\widehat{f}$ from $\mathbb{Z}_{>0}$ to $\mathbb{Z}_{p}$ denoted by
$\beta$ and $\widehat{\beta},$ respectively. We write the value of
$\beta$(resp. $\widehat{\beta}$) at $\lambda\in\mathbb{Z}_{p}$ by
$\beta_{\lambda}$(resp. $\widehat{\beta}_{\lambda}$). The following lemma
gives a relation between $\beta$ and $\widehat{\beta}.$

######  Lemma 3.2.

Let $\lambda\in\mathbb{Z}_{p}$ and
$a\in\mathbb{Z}_{p}\backslash\mathbb{Z}_{\leq 0},$ then we have

$\beta_{\lambda}+\widehat{\beta}_{-\lambda-a}=0.$

Before the proof of Lemma 3.2, we need to introduce some notation and one more
lemma.

######  Definition 3.3.

For $\alpha\in\mathbb{Z}_{p}$ and $n\in\mathbb{Z}_{\geq 1},$ let us define

$\\{\alpha\\}_{n}=\prod_{\begin{subarray}{c}1\leq i\leq n\\\
p\nmid(\alpha+i-1)\end{subarray}}(\alpha+i-1)$

and $\\{\alpha\\}_{0}=1.$

######  Lemma 3.4.

Let $a\in\mathbb{Z}_{p}\backslash\mathbb{Z}_{\leq 0}$ and let
$x,y\in\mathbb{Z}_{\geq 0}$ satisfying $x+y+a\equiv 0\mod p^{n}.$ Then

$(-1)^{f_{x}}\frac{\\{1\\}_{x}}{\\{a\\}_{x}}\equiv(-1)^{f_{y}}\frac{\\{1\\}_{y}}{\\{a\\}_{y}}\mod
p^{n}.$

where $f_{x}=\ell_{x}-\lfloor\ell_{x}/p\rfloor,$
$f_{y}=\ell_{y}-\lfloor\ell_{y}/p\rfloor$ and
$\ell_{x},\ell_{y}\in\\{0,1,\cdots,q-1\\}$ such that $x-\ell_{x}\equiv
y-\ell_{y}\equiv 0\mod q.$

###### Proof.

For simplicity, we write $[\quad]_{p}$ for the product that those terms
divided by $p$ are omitted and note that if we have $a+x+y\equiv 0\mod{p^{n}}$
and

$(-1)^{f_{x}}\frac{\\{1\\}_{x}}{\\{a\\}_{x}}\equiv(-1)^{f_{y}}\frac{\\{1\\}_{y}}{\\{a\\}_{y}}\mod
p^{n},$

then

$(-1)^{f_{x}}\frac{\\{1\\}_{x+q}}{\\{a\\}_{x+q}}\equiv(-1)^{f_{y}}\frac{\\{1\\}_{y-q}}{\\{a\\}_{y-q}}\mod
p^{n}.$ (3.1)

Case I : $p\neq 2$

First we prove that the lemma is true for the following two cases.

(1) The case that there exist $z,\ell_{x},\ell_{y}$ with $z\in\mathbb{Z}_{\geq
0},p|z$ and $0\leq\ell_{x},\ell_{y}\leq p-1$ such that

$\left\\{\begin{aligned} x&=z+\ell_{x}\\\ y&=z+\ell_{y}.\end{aligned}\right.$

(2) The case that there exist $z,\ell_{x},\ell_{y}$ with $z\in\mathbb{Z}_{\geq
0},p|z$ and $0\leq\ell_{x},\ell_{y}\leq p-1$ such that

$\left\\{\begin{aligned} &x=z+\ell_{x}\\\
&y=z+p+\ell_{y}.\end{aligned}\right.$

Note that we can assume that $\ell_{x}\leq\ell_{y}$, since (1) the case
$\ell_{y}<\ell_{x}$ follows from symmetry and for (2) the case
$\ell_{y}<\ell_{x}$ follows from using (3.1).

Then in case $(1)$, one has

$\displaystyle\frac{\\{1\\}_{z+\ell_{y}}}{\\{a\\}_{z+\ell_{y}}}\bigg{/}\frac{\\{1\\}_{z+\ell_{x}}}{\\{a\\}_{z+\ell_{x}}}$
$\displaystyle=$
$\displaystyle\frac{(1+z+\ell_{x})\cdots(1+z+\ell_{y}-1)}{(a+z+\ell_{x})\cdots(a+z+\ell_{y}-1)}$
$\displaystyle\equiv$
$\displaystyle\frac{(1+z+\ell_{x})\cdots(1+z+\ell_{y}-1)}{(-z-\ell_{y})\cdots(-z-\ell_{x}-1)}\mod
p^{n}$ $\displaystyle=$ $\displaystyle(-1)^{\ell_{y}-\ell_{x}}.$

For case $(2)$, we have

$\displaystyle\frac{\\{1\\}_{z+p+\ell_{y}}}{\\{a\\}_{z+p+\ell_{y}}}\bigg{/}\frac{\\{1\\}_{z+\ell_{x}}}{\\{a\\}_{z+\ell_{x}}}$
$\displaystyle=$
$\displaystyle\bigg{[}\frac{(1+z+\ell_{x})\cdots(z+p+\ell_{y})}{(a+z+\ell_{x})\cdots(a+z+p+\ell_{y}-1)}\bigg{]}_{p}$
$\displaystyle\equiv$
$\displaystyle\bigg{[}\frac{(1+z+\ell_{x})\cdots(z+p+\ell_{y})}{(-z-p-\ell_{y})\cdots(-1-z-\ell_{x})}\bigg{]}_{p}\mod
p^{n}$ $\displaystyle=(-1)^{p-1+\ell_{y}-\ell_{x}}=(-1)^{\ell_{y}-\ell_{x}}.$

One can show that for general $x,y$ can be written in the form of

$\left\\{\begin{aligned} &x=kp+z+\ell_{x}\\\
&y=-kp+z+\ell_{y}\end{aligned}\right.$

or

$\left\\{\begin{aligned} &x=kp+z+\ell_{x}\\\
&y=-kp+z+p+\ell_{y}\end{aligned}\right.$

with $k\in\mathbb{Z}$ and $z\in\mathbb{Z}_{\geq 0},p|z.$ Then we can use case
(1), (2) (maybe the role of $x,y$ change depending on $\ell_{x},\ell_{y}$) and
(3.1) inductively to prove

$(-1)^{f_{x}}\frac{\\{1\\}_{x}}{\\{a\\}_{x}}\equiv(-1)^{f_{y}}\frac{\\{1\\}_{y}}{\\{a\\}_{y}}\mod
p^{n}.$

Case II : $p=2$

The proof is similar. We first prove this lemma for the cases:

(1) The case that there exist $z,\ell_{x},\ell_{y}$ with
$4|z,z\in\mathbb{Z}_{\geq 0}$ and $0\leq\ell_{x},\ell_{y}\leq 3$ such that

$\left\\{\begin{aligned} x&=z+\ell_{x}\\\ y&=z+\ell_{y}.\end{aligned}\right.$

(2) The case that there exist $z,\ell_{x},\ell_{y}$ with
$4|z,z\in\mathbb{Z}_{\geq 0}$ and $0\leq\ell_{x},\ell_{y}\leq 3$ such that

$\left\\{\begin{aligned} &x=z+\ell_{x}\\\
&y=z+4+\ell_{y}.\end{aligned}\right.$

Then we show that for any $x,y$ such that $x+y+a\equiv 0\mod{2^{n}},$ there
exist $z\in\mathbb{Z}_{\geq 0},4|z,k\in\mathbb{Z}$ such that

$\left\\{\begin{aligned} &x=4k+z+\ell_{x}\\\
&y=-4k+z+\ell_{y}\end{aligned}\right.$

or

$\left\\{\begin{aligned} &x=4k+z+\ell_{x}\\\
&y=-4k+z+4+\ell_{y}.\end{aligned}\right.$

Again, by (3.1), the result follows. ∎

Now we can prove Lemma 3.2.

###### (proof of Lemma 3.2).

It suffices to prove

$\beta_{\lambda}+\widehat{\beta}_{-\lambda-a}\equiv 0\mod p^{n}$

for all $n\in\mathbb{Z}_{\geq 0}.$ Since
$\beta_{\lambda}\equiv\beta_{\lambda^{\prime}}$ and
$\widehat{\beta}_{\lambda}\equiv\widehat{\beta}_{\lambda^{\prime}}\mod p^{n}$
if $\lambda\equiv\lambda^{\prime}\mod p^{n}$, we can pick
$b\in\mathbb{Z}_{>0}$ such that $b\equiv\lambda\mod p^{n}$ and consider
$\beta_{b}+\widehat{\beta}_{-b-a}$ instead.

If $p\nmid b$, then $\beta_{b}=1/b$ and $\widehat{\beta}_{-b-a}=-1/b.$ Hence,
$\beta_{b}+\widehat{\beta}_{-b-a}=0.$ Now we suppose $p|b.$ Pick
$x,y\in\mathbb{Z}_{>0}$ and $0<x,y\leq p^{2n}.$ such that $x\equiv b\mod
p^{n}$ but $p^{n+1}\nmid x$ and $x+y+a\equiv 0\mod p^{2n}.$ Then we claim

$\beta_{x}+\widehat{\beta}_{y}\equiv 0\mod p^{n}.$

By [1, Lemma 3.6], we have

$\frac{A^{(1)}_{x/p}}{A_{x}}=\bigg{(}\frac{\\{1\\}_{x}}{\\{a\\}_{x}}\bigg{)}^{s},\quad\frac{A^{(1)}_{\frac{y-l}{p}}}{A_{y}}=\bigg{(}\frac{\\{1\\}_{y}}{\\{a\\}_{y}}\bigg{)}^{s}.$

Therefore $\beta_{x}+\widehat{\beta}_{y}$ becomes

$\frac{1-c^{x/p}(\frac{\\{1\\}_{x}}{\\{a\\}_{x}})^{s}}{x}+\frac{1-(-1)^{se}c^{-\frac{y+a}{p}}(\frac{\\{1\\}_{y}}{\\{a\\}_{y}})^{s}}{y+a}.$
(3.2)

Since $x+y+a\equiv 0\mod p^{2n},$ we write $x+y+a=zp^{2n}$ for some
$z\in\mathbb{Z}_{p}.$ Since $p^{n+1}\nmid x$, we have ${\rm ord}_{p}(y+a)={\rm
ord}_{p}(-x+zp^{2n})\leq n.$ Therefore after modulo $p^{n},$ we have

$\displaystyle\frac{1-c^{x/p}(\frac{\\{1\\}_{x}}{\\{a\\}_{x}})^{s}}{x}+\frac{1-(-1)^{se}c^{-\frac{y+a}{p}}(\frac{\\{1\\}_{y}}{\\{a\\}_{y}})^{s}}{y+a}$
(3.3) $\displaystyle\equiv$
$\displaystyle\frac{1-c^{x/p}(\frac{\\{1\\}_{x}}{\\{a\\}_{x}})^{s}}{x}+\frac{1-(-1)^{se}c^{-\frac{y+a}{p}}(\frac{\\{1\\}_{y}}{\\{a\\}_{y}})^{s}}{-x}\mod{p^{n}}$
$\displaystyle=$
$\displaystyle\frac{-c^{x/p}(\frac{\\{1\\}_{x}}{\\{a\\}_{x}})^{s}+c^{-\frac{y+a}{p}}((-1)^{e}\frac{\\{1\\}_{y}}{\\{a\\}_{y}})^{s}}{x}.$

Observe that $c^{-\frac{y+a}{p}}=c^{\frac{x}{p}-zp^{2n-1}}\equiv c^{x/p}\mod
p^{2n}.$ Thus, in order to prove equation (3.3) is $0$ modulo $p^{n}$, it
suffices to prove that

$\frac{\\{1\\}_{x}}{\\{a\\}_{x}}\equiv(-1)^{e}\frac{\\{1\\}_{y}}{\\{a\\}_{y}}\mod
p^{2n}.$

And this follows from Lemma 3.4. Therefore the result follows. ∎

##  4 Transformation Formulas

In this section, we will recall two conjectures of $p$-adic hypergeometric
functions. The first one is “Transformation Formulas between $p$-adic
hypergeometric functions $\widehat{\mathscr{F}}_{a,...,a}^{(\sigma)}(t)$ and
$p$-adic hypergeometric functions of logarithmic type” and the second one is
“Transformation formula on Dwork’s $p$-adic hypergeometric Functions”. Then we
prove the second conjecture implies the first conjecture.

###  4.1 Transformation Formula of $\mathscr{F}_{a}^{\;(\sigma)}(t)$ and
$\widehat{\mathscr{F}}_{a}^{\;(\widehat{\sigma})}(t)$

######  Conjecture 4.1 ([4, Conjecture 4.12], Transformation Formula between
$\mathscr{F}_{a}^{\;(\sigma)}(t)$ and
$\widehat{\mathscr{F}}_{a}^{\;(\widehat{\sigma})}(t)$).

Let $\sigma(t)=ct^{p}$ and $\widehat{\sigma}(t)=c^{-1}t^{p}$. Let
$a\in\mathbb{Z}_{p}\backslash\mathbb{Z}_{\leq 0}$ and the $r$th Dwork prime
$a^{(r)}=a$ for some $r>0.$ Put
$h(t):=\prod_{i=0}^{r-1}F_{a^{(i)},\cdots,a^{(i)}}(t)_{<p},$ then

$\mathscr{F}_{a}^{\;(\sigma)}(t)=-\widehat{\mathscr{F}}_{a}^{\;(\widehat{\sigma})}(t^{-1})$

in the ring $W\langle t,t^{-1},h(t)^{-1}\rangle$ where
$\widehat{\mathscr{F}}_{a}^{\;(\widehat{\sigma})}(t^{-1})$ is defined as
$\omega(\widehat{\mathscr{F}}_{a}^{\;(\widehat{\sigma})}(t)).$ The map

$\omega:W\langle t,t^{-1},h(t)^{-1}\rangle\rightarrow W\langle
t,t^{-1},h(t)^{-1}\rangle$

is defined in [4, §4.3, Proposition 4.11] and
$\mathscr{F}_{a}^{\;(\sigma)}(t)$ is the $p$-adic hypergeometric functions of
logarithmic type.

######  Remark 4.2.

From [4, Theorem 4.14], we know that the conjecture is true for
$s=2,a\in\frac{1}{N}\mathbb{Z},0<a<1$ and $p>N.$

###  4.2 Transformation formula on Dwork’s $p$-adic Hypergeometric Functions

First, let us recall the definition of Dwork’s $p$-adic hypergeometric
functions.

######  Definition 4.3 (Dwork’s $p$-adic hypergeometric functions).

Let $s\geq 1$ be an integer. For $(a_{1},\cdots,a_{s})\in\mathbb{Z}_{p}^{s}$,
Dwork’s $p$-adic hypergeometric function is defined as

$\mathscr{F}^{{\rm
Dw}}_{a_{1},\cdots,a_{s}}(t):=F_{a_{1},\cdots,a_{s}}(t)/F_{a_{1}^{\prime},\cdots,a_{s}^{\prime}}(t^{p})$

where $F_{a_{1},\cdots,a_{s}}(t)$ and
$F_{a_{1}^{\prime},\cdots,a_{s}^{\prime}}(t)$ are hypergeometric power series.

######  Conjecture 4.4 ([4, Conjecture 4.18],Transformation formula on Dwork’s
$p$-adic Hypergeometric Functions).

Suppose $a_{1}=\cdots=a_{s}=a.$ Let $l$ is the unique integer in
$\\{0,1,\cdots,p-1\\}$ such that $a+l\equiv 0\mod p.$ Then

$\mathscr{F}^{{\rm Dw}}_{a}(t)=((-1)^{s}t)^{l}\mathscr{F}^{{\rm
Dw}}_{a}(t^{-1}).$

######  Remark 4.5.

From [4, Theorem 4.20], we know that the conjecture is true for
$s=2,a\in\frac{1}{N}\mathbb{Z},0<a<1$ and $p>N.$

In fact, we have a special case $s=1$ of Conjecture 4.4 for $p$ is odd. For
$p=2,$ we find a counterexample, but it only differs by a minus sign.

######  Theorem 4.6.

Let $p$ be an odd prime and $a\in\mathbb{Z}_{p},$ then we have

$\mathscr{F}^{{\rm Dw}}_{a}(t)=(-t)^{l}\mathscr{F}^{{\rm Dw}}_{a}(t^{-1}),$

where $l$ is the unique integer in $\\{0,1,\cdots,p-1\\}$ such that $a+l\equiv
0\mod p.$

For $p=2$, we have

$\mathscr{F}^{{\rm Dw}}_{a}(t)=\pm(-t)^{l}\mathscr{F}^{{\rm Dw}}_{a}(t^{-1}).$

In particular, we have a counterexample for Conjecture 4.4.

###### Proof.

By [2, p.37, Theorem 2], we have

$\mathscr{F}^{\rm
Dw}_{a}(t)\equiv\frac{F_{a}(t)_{<p^{n}}}{F_{a^{\prime}}(t^{p})_{<p^{n}}}=\frac{(1-t)^{-a}_{<p^{n}}}{(1-t^{p})^{-a^{\prime}}_{<p^{n}}}\equiv\frac{(1-t)^{-N}_{<p^{n}}}{(1-t^{p})^{-N^{\prime}}_{<p^{n}}}\mod
p^{n}$

for some $N\in\mathbb{Z}_{>0}$ and $a\equiv N\mod p.$

Then we obtain

$\displaystyle\mathscr{F}^{\rm Dw}_{a}(t^{-1})$
$\displaystyle\equiv\left.\frac{(1-t)^{-N}_{<p^{n}}}{(1-t^{p})^{-N^{\prime}}_{<p^{n}}}\right|_{t^{-1}}\mod
p^{n}$
$\displaystyle\equiv\left.\frac{(1-t)^{-N}}{(1-t^{p})^{-N^{\prime}}}\right|_{t^{-1}}$
$\displaystyle=\frac{(-t)^{-pN^{\prime}}}{(-t)^{-pN^{\prime}}}\cdot\frac{(1-t^{-1})^{-N}}{(1-t^{-p})^{-N^{\prime}}}$
$\displaystyle=\frac{(-t)^{-l}(1-t)^{-N}}{(1-t^{p})^{-N^{\prime}}}$

when $p$ is odd. Therefore the result follows.

For $p=2,$ we have

$\displaystyle\mathscr{F}^{\rm Dw}_{a}(t^{-1})$
$\displaystyle\equiv=\pm\frac{(-t)^{-l}(1-t)^{-N}}{(1-t^{2})^{-N^{\prime}}}\mod
2^{n}.$

For example, if we take $a=1,$ then

$\mathscr{F}^{\rm Dw}_{a}(t^{-1})=1+t$

but

$(-t)\mathscr{F}^{\rm Dw}_{a}(t^{-1})=-(1+t).$

∎

So we propose another conjecture about the transformation formula of Dwork
$p$-adic hypergeometric functions to include Theorem 4.6 as a special case.
And we will find that our new conjecture is enough for us to derive our main
Theorem 4.9.

######  Conjecture 4.7 (Transformation formula on Dwork’s $p$-adic
Hypergeometric Functions).

Suppose $a_{1}=\cdots=a_{s}=a.$ Let $l$ is the unique integer in
$\\{0,1,\cdots,p-1\\}$ such that $a+l\equiv 0\mod p.$ Then for odd prime $p,$
we have

$\mathscr{F}^{{\rm Dw}}_{a}(t)=((-1)^{s}t)^{l}\mathscr{F}^{{\rm
Dw}}_{a}(t^{-1}).$

For $p=2,$ we have

$\mathscr{F}^{{\rm Dw}}_{a}(t)=\pm((-1)^{s}t)^{l}\mathscr{F}^{{\rm
Dw}}_{a}(t^{-1}).$

###  4.3 The proof of Conjecture 4.7 implies Conjecture 4.1

First, we prove the following lemma.

######  Lemma 4.8.

Let $a\in\mathbb{Z}_{p},$ $m,k,d\in\mathbb{Z}_{\geq 0}$ with $0\leq m\leq
p^{n}-1$ and $0\leq d\leq n.$ If Conjecture 4.7 is true, then we have

$\sum_{\begin{subarray}{c}i\equiv k\mod{p^{n-d}}\\\ 0\leq i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n-d}\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\equiv
0\mod p^{d+1}.$ (4.1)

###### Proof.

The proof of this lemma is similar to [1, Lemma 3.12].

Put

$F^{(r)}(t):=\sum_{i=0}^{\infty}A_{i}^{(r)}t^{i}$

and

$F_{k}^{(r)}(t):=\sum_{i\equiv k\mod
p^{n-d}}A_{i}^{(r)}t^{i}=p^{-n+d}\sum_{u=0}^{p^{n-d}-1}\zeta^{-ku}F(\zeta^{u}t)$

where $\zeta$ is a primitive $p^{n-d}$ root of unity. For simplicity, we write
$F(t)$ and $F_{k}(t)$ for $F^{(0)}(t)$ and $F_{k}^{(0)}(t),$ respectively.

Since we suppose Conjecture 4.7 is true, we have

$\displaystyle\frac{F(t)}{F^{(n-d)}(t^{p^{n-d}})}$ $\displaystyle=$
$\displaystyle\frac{F(t)}{F^{(1)}(t^{p})}\frac{F^{(1)}(t^{p})}{F^{(2)}(t^{p^{2}})}\cdots\frac{F^{(n-d-1)}(t^{p^{n-d-1}})}{F^{(n-d)}(t^{p^{n-d}})}$
$\displaystyle=$
$\displaystyle\pm((-1)^{s}t)^{l_{0}}\left.\frac{F(t)}{F^{(1)}(t^{p})}\right|_{t^{-1}}\cdot(\pm)((-1)^{s}t^{p})^{l_{1}}\left.\frac{F^{(1)}(t^{p})}{F^{(2)}(t^{p^{2}})}\right|_{t^{-1}}$
$\displaystyle\quad\cdots(\pm)((-1)^{s}t^{p^{n-d-1}})^{l_{n-d-1}}\left.\frac{F^{(n-d-1)}(t^{p^{n-d-1}})}{F^{(n-d)}(t^{p^{n-d}})}\right|_{t^{-1}}$
$\displaystyle=$ $\displaystyle(\pm
1)^{n-d}(-1)^{s(l_{0}+l_{1}+\cdots+l_{n-d-1})}t^{l_{0}+l_{1}p+\cdots+l_{n-d-1}p^{n-l-1}}\left.\frac{F(t)}{F^{(1)}(t^{p})}\right|_{t^{-1}}\cdots\left.\frac{F^{(n-d-1)}(t^{p^{n-d-1}})}{F^{(n-d)}(t^{p^{n-d}})}\right|_{t^{-1}}$

where $l_{0},l_{1},\cdots,l_{n-d-1}$ are the numbers in $\\{0,1,\cdots,p-1\\}$
such that $a+l_{0}\equiv a^{\prime}+l_{1}\equiv\cdots\equiv
a^{(n-d-1)}+l_{n-d-1}\equiv 0\mod p.$

From the Dwork congruence [2, p.37, Theorem 2], we have

$\frac{F^{(i)}(t)}{F^{(i+1)}(t^{p})}\equiv\frac{F^{(i)}(t)_{<p^{m}}}{F^{(i+1)}(t^{p})_{<p^{m}}}\mod
p^{n}$

when $m\geq n\geq 1.$ Therefore one has

$\frac{F^{(i)}(t^{p})}{F^{(i+1)}(t^{p^{2}})}\equiv\frac{F^{(i)}(t^{p})_{<p^{n+1}}}{F^{(i+1)}(t^{p^{2}})_{<p^{n+1}}}\mod
p^{n},\frac{F^{(i)}(t^{p^{2}})}{F^{(i+1)}(t^{p^{3}})}\equiv\frac{F^{(i)}(t^{p^{2}})_{<p^{n+2}}}{F^{(i+1)}(t^{p^{3}})_{<p^{n+2}}}\mod
p^{n},\cdots$

After modulo $p^{d+1}$ and writing
$s^{\prime}=s(l_{0}+l_{1}+\cdots+l_{n-d-1})$ and
$l^{\prime}=l_{0}+l_{1}p+\cdots+l_{n-d-1}p^{n-d-1},$ we have

$\displaystyle\frac{F(t)}{F^{(n-d)}(t^{p^{n-d}})}$ $\displaystyle\equiv(\pm
1)^{n-d}(-1)^{s^{\prime}}t^{l^{\prime}}\left.\frac{F(t)_{<p^{n}}}{F^{(1)}(t^{p})_{<p^{n}}}\right|_{t^{-1}}\cdots\left.\frac{F^{(n-d-1)}(t^{p^{n-d-1}})_{<p^{n}}}{F^{(n-d)}(t^{p^{n-d}})_{<p^{n}}}\right|_{t^{-1}}\mod
p^{d+1}\mathbb{Z}_{p}[[t]]$ $\displaystyle=(\pm
1)^{n-d}(-1)^{s^{\prime}}t^{l^{\prime}}\left.\frac{F(t)_{<p^{n}}}{F^{(n-d)}(t^{p^{n-d}})_{<p^{n}}}\right|_{t^{-1}}.$

Therefore

$\frac{F(t)}{F^{(n-d)}(t^{p^{n-d}})}=(\pm
1)^{n-d}(-1)^{s^{\prime}}t^{l^{\prime}}\left.\frac{F(t)_{<p^{n}}}{F^{(n-d)}(t^{p^{n-d}})_{<p^{n}}}\right|_{t^{-1}}+p^{d+1}\sum_{i=0}^{\infty}a_{i}t^{i}$
(4.2)

where $a_{i}\in\mathbb{Z}_{p}.$

If we substitute $\zeta^{u}t$ for $t,$ we have

$\displaystyle\frac{F(\zeta^{u}t)}{F^{(n-d)}(t^{p^{n-d}})}$
$\displaystyle=(\pm
1)^{n-d}(-1)^{s^{\prime}}(\zeta^{u}t)^{l^{\prime}}\left.\frac{F(t)_{<p^{n}}}{F^{(n-d)}(t^{p^{n-d}})_{<p^{n}}}\right|_{\zeta^{-u}t^{-1}}+p^{d+1}\sum_{i=0}^{\infty}a_{i}\zeta^{ui}t^{i}$
(4.3) $\displaystyle=(\pm
1)^{n-d}(-1)^{s^{\prime}}(\zeta^{u}t)^{l^{\prime}}\left.\frac{F(\zeta^{-u}t)_{<p^{n}}}{F^{(n-d)}(t^{p^{n-d}})_{<p^{n}}}\right|_{t^{-1}}+p^{d+1}\sum_{i=0}^{\infty}a_{i}\zeta^{ui}t^{i}.$

Dividing equation (4.3) by equation (4.2) and expanding the quotient, one has

$F(\zeta^{u}t)\left.F(t)_{<p^{n}}\right|_{t^{-1}}\cdot
t^{p^{n}-1}-F(t)F(\left.\zeta^{-u}t)_{<p^{n}}\right|_{t^{-1}}\cdot{t^{p^{n}-1}}\cdot\zeta^{ul^{\prime}}=p^{d+1}\sum_{i=0}^{\infty}b_{i}(\zeta^{u})t^{i}$

where $b_{i}(x)\in\mathbb{Z}_{p}[x]$ are polynomials which does not depend on
$u.$

Taking $\sum\limits_{u=0}^{p^{n-d}-1}\zeta^{-uk}(-)$ on both side and noting
that $l^{\prime}\in\\{0,1,\cdots,p^{n-d}-1\\}$ is the number such that
$a+l^{\prime}\equiv 0\mod p^{n-d}$, one has

$p^{n-d}\bigg{(}F_{k}(t)\left.F(t)_{<p^{n}}\right|_{t^{-1}}\cdot
t^{p^{n}-1}-F(t)F_{-k-a}(\left.t)_{<p^{n}}\right|_{t^{-1}}\cdot{t^{p^{n}-1}}\bigg{)}=p^{n-d}\cdot
p^{d+1}g(x).$

for some power series $g(x).$

Therefore

$\frac{F_{k}(t)}{F(t)}\equiv\left.\frac{F_{-k-a}(t)_{<p^{n}}}{F(t)_{<p^{n}}}\right|_{t^{-1}}\mod
p^{d+1}.$

On the other hand, using a similar proof, one has

$\frac{F_{k}(t)}{F(t)}\equiv\frac{F_{k}(t)_{<p^{n}}}{F(t)_{<p^{n}}}\mod
p^{d+1},$

(cf. the last paragraph in the proof of [1, Lemma 3.12]) so we conclude

$\frac{F_{k}(t)_{<p^{n}}}{F(t)_{<p^{n}}}\equiv\left.\frac{F_{-k-a}(t)_{<p^{n}}}{F(t)_{<p^{n}}}\right|_{t^{-1}}\mod
p^{d+1}.$ (4.4)

Using (4.4), one has

$F_{k}(t)\left.F(t)_{<p^{n}}\right|_{t^{-1}}\cdot
t^{p^{n}-1}=F(t)F_{-k-a}(\left.t)_{<p^{n}}\right|_{t^{-1}}\cdot{t^{p^{n}-1}}\mod
p^{d+1}.$ (4.5)

Then this lemma follows from comparing the coefficients of $t^{m}$ of (4.5). ∎

######  Theorem 4.9.

Conjecture 4.7 implies Conjecture 4.1.

###### Proof.

By congruence relations of $\mathscr{F}^{(\sigma)}_{a}(t)$ and
$\widehat{\mathscr{F}}^{\;(\widehat{\sigma})}_{a}(t)$ (Theorem 2.4 and Theorem
2.5), Conjecture 4.1 is equivalent to the statement

$\frac{G^{(\sigma)}_{a}(t)_{<p^{n}}}{F_{a}(t)_{<p^{n}}}\equiv-\left.\frac{\widehat{G}^{\;(\widehat{\sigma})}_{a}(t)_{<p^{n}}}{F_{a}(t)_{<p^{n}}}\right|_{t^{-1}}\mod
p^{n}W[[t]]$

for all $n\in\mathbb{Z}_{\geq 0}.$ So it suffices to show that

$\sum_{\begin{subarray}{c}i+j=m\\\ 0\leq i,j\leq
p^{n}-1\end{subarray}}B_{i}A_{p^{n}-j-1}+\widehat{B}_{p^{n}-j-1}A_{i}\equiv
0\mod p^{n}.$

for any $m$ with $0\leq m\leq 2(p^{n}-1).$

First, we suppose $0\leq m\leq p^{n}-1.$ Using the relation
$\beta_{k}=B_{k}/A_{k}$ and $\widehat{\beta}_{k}=\widehat{B}_{k}/A_{k}$ when
$k\in\mathbb{Z}_{\geq 0},$ we write

$\displaystyle\sum_{\begin{subarray}{c}i+j=m\\\ 0\leq i,j\leq
m\end{subarray}}B_{i}A_{p^{n}-j-1}+\widehat{B}_{p^{n}-j-1}A_{i}$ (4.6)
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}i+j=m\\\ 0\leq i,j\leq
m\end{subarray}}\beta_{i}A_{i}A_{p^{n}-j-1}+\sum_{\begin{subarray}{c}i^{\prime}+j^{\prime}=m\\\
0\leq i^{\prime},j^{\prime}\leq
m\end{subarray}}\widehat{\beta}_{p^{n}-j^{\prime}-1}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{p^{n}-1}\bigg{[}\sum_{\begin{subarray}{c}i\equiv
k\mod p^{n}\\\ 0\leq i\leq
m\end{subarray}}\beta_{i}A_{i}A_{p^{n}-j-1}+\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n}\\\ 0\leq j^{\prime}\leq
m\end{subarray}}\widehat{\beta}_{p^{n}-j^{\prime}-1}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\bigg{]}$
$\displaystyle\equiv$
$\displaystyle\sum_{k=0}^{p^{n}-1}\beta_{k}\bigg{[}\sum_{\begin{subarray}{c}i\equiv
k\mod p^{n}\\\ 0\leq i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n}\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\bigg{]}\mod{p^{n}}.$

By Lemma 4.8, we have

$\sum_{\begin{subarray}{c}i\equiv k\mod p^{n}\\\ 0\leq i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n}\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\equiv
0\mod p.$

So we can rewrite the equation (4.6) as

$\sum_{k^{\prime}=0}^{p^{n-1}-1}\sum_{k\equiv k^{\prime}\mod
p^{n-1}}\beta_{k}\bigg{[}\sum_{\begin{subarray}{c}i\equiv k\mod p^{n}\\\ 0\leq
i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n}\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\bigg{]}$

and this equals

$\displaystyle\sum_{k^{\prime}=0}^{p^{n-1}-1}\beta_{k^{\prime}}\sum_{k\equiv
k^{\prime}\mod p^{n-1}}\bigg{[}\sum_{\begin{subarray}{c}i\equiv k\mod p^{n}\\\
0\leq i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n}\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\bigg{]}$
$\displaystyle\equiv\sum_{k^{\prime}=0}^{p^{n-1}-1}\beta_{k^{\prime}}\bigg{[}\sum_{\begin{subarray}{c}i\equiv
k\mod p^{n-1}\\\ 0\leq i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n-1}\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\bigg{]}\mod
p^{n}.$

Again by Lemma 4.8 and using the same argument, the equation equals

$\sum_{k^{\prime\prime}=0}^{p^{n-2}-1}\beta_{k^{\prime\prime}}\bigg{[}\sum_{\begin{subarray}{c}i\equiv
k\mod p^{n-2}\\\ 0\leq i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p^{n-2}\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\bigg{]}\mod
p^{n}.$

Continuing this process, we finally get the equation equals

$\sum_{k=0}^{p-1}\beta_{k}\bigg{[}\sum_{\begin{subarray}{c}i\equiv k\mod p\\\
0\leq i\leq m\\\
i+j=m\end{subarray}}A_{i}A_{p^{n}-j-1}-\sum_{\begin{subarray}{c}p^{n}-j^{\prime}-1\equiv-
k-a\mod p\\\ 0\leq j^{\prime}\leq m\\\
i^{\prime}+j^{\prime}=m\end{subarray}}A_{i^{\prime}}A_{p^{n}-j^{\prime}-1}\bigg{]}\equiv
0\mod p^{n}.$

Thus

$\sum_{\begin{subarray}{c}i+j=m\\\ 0\leq i,j\leq
m\end{subarray}}B_{i}A_{p^{n}-j-1}+\widehat{B}_{p^{n}-j-1}A_{i}\equiv 0\mod
p^{n}.$

On the other hand, if $p^{n}-1\leq m\leq 2(p^{n}-1)$, we put
$i^{\prime}=p^{n}-i-1$ and $j^{\prime}=p^{n}-j-1$. Then

$\sum_{\begin{subarray}{c}i+j=m\\\ 0\leq i,j\leq
p^{n}-1\end{subarray}}B_{i}A_{p^{n}-j-1}+\widehat{B}_{p^{n}-j-1}A_{i}$

becomes

$\sum_{\begin{subarray}{c}i^{\prime}+j^{\prime}=2(p^{n}-1)-m\\\ 0\leq
i^{\prime},j^{\prime}\leq
p^{n}-1\end{subarray}}B_{p^{n}-i^{\prime}-1}A_{j^{\prime}}+\widehat{B}_{j^{\prime}}A_{p^{n}-i^{\prime}-1}.$

Let $m^{\prime}=2(p^{n}-1)-m$ and interchange the role of $B_{k}$ and
$\widehat{B}_{k}$, we find this is just the case above. Therefore the result
follows. ∎

######  Corollary 4.10.

Conjecture 4.1 is true when $s=1$.

###### Proof.

This follows from Theorem 4.9 and Theorem 4.6. ∎

## References

  * [1] Asakura, M.: New p-adic hypergeometric functions and syntomic regulators. arXiv:1811.03770.
  * [2] Dwork, B.: p-adic cycles. Publ. Math. IHES, tome 37 (1969), 27-115.
  * [3] L. J. Slater:  Generalized hypergeometric functions, Cambridge Univ. Press, Cambridge (1966).
  * [4] Wang Chung-Hsuan.: Congruence relations for $p$-adic hypergeometric functions $\widehat{\mathscr{F}}_{a,...,a}^{(\sigma)}(t)$ and its transformation formula arXiv:2001.08117.

Department of Mathematics, National Cheng Kung University
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