# Frobenius intertwiners for $q$-difference equations

Andrey Smirnov⋆

###### Abstract.

In this note we consider a class of $q$-hypergeometric equations describing
the quantum difference equation for $X=T^{*}\mathbb{P}^{1}$ . We show that
over $\mathbb{Q}_{p}$ these equations are equipped with the Frobenius action
$z\to z^{p}$. We obtain an explicit formula for the constant term of the
Frobenius intertwiner in therms of $p$ \- adic $q$ \- gamma function of
Koblitz. In the limit $q\to 1$ our formula degenerates to the Frobenius
intertwiners for the hypergeometric differential equations discovered by
B.Dwork.

Department of Mathematics, University of North Carolina at Chapel Hill

Chapel Hill, NC 27599-3250, USA

Key words: Frobenius intertwiners, $q$ \- hypergeometric functions, $p$ \-
adic $q$ \- Gamma function

2020 Mathematics Subject Classification: 14G33 (11D79, 32G34, 33C05, 33E30)

††footnotetext: ${}^{\star}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>
##  1\. q-hypergeometric functions

In this section we let $\mathbb{K}=\mathbb{C}$. In the next section we switch
to the field of $p$-adic numbers.

### 1.1.

Let $D$ be the $q$-shift operator acting by $Df(z)=f(zq)$. We consider a
$q$-holonomic module $\mathcal{H}({\boldsymbol{u}},\hbar,q,z)$ with
$\mathbb{K}^{\times}$-valued parameters
${\boldsymbol{u}}=(u_{1},\dots,u_{n})$, $\hbar$ given by the following
$q$-difference equation

(1.1) $\displaystyle\Big{(}\prod\limits_{i=1}^{n}\dfrac{1-u^{-1}_{i}D}{1-\hbar
q^{-1}u^{-1}_{i}D}\Big{)}F(z)=zF(z).$

This $q$-difference equation can be written in a more standard form:

(1.2) $\displaystyle P(\boldsymbol{a},\hbar,z)F(z)=0,$

where $P(\boldsymbol{a},\hbar,z)$ denotes the $q$-difference operator:

(1.3) $\displaystyle
P(\boldsymbol{a},\hbar,z)=\sum_{m=0}^{n}\,\alpha_{m}(1-z\hbar^{m})D^{m}.$

and $\alpha_{m}$ denotes the coefficient of $x^{m}$ in
$\prod\limits_{i=1}^{n}(1-x/u_{i})$.

### 1.2.

Let us consider the $q$-hypergeometric power series

(1.4) $\displaystyle
F_{i}({\boldsymbol{u}},\hbar,q,z)=\sum\limits_{d=0}^{\infty}\,\Big{(}\prod\limits_{j=1}^{n}\,\dfrac{(u_{i}\hbar/u_{j})_{d}}{(u_{i}q/u_{j})_{d}}\Big{)}\,z^{d}.$

Set $e_{i}(z)=\exp\Big{(}{\frac{\ln(z)\ln(u_{i})}{\ln(q)}}\Big{)},$ so that
$e_{i}(zq)=e_{i}(z)u_{i}$. One verifies that the functions

(1.5)
$\displaystyle\tilde{F}_{i}({\boldsymbol{u}},\hbar,q,z)=e_{i}(z)F_{i}({\boldsymbol{u}},\hbar,q,z)$

satisfy (1.1) for all $i=1,\dots,n$. Thus,
$\tilde{F}_{i}({\boldsymbol{u}},\hbar,q,z)$ give a basis of solutions for
$\mathcal{H}({\boldsymbol{u}},\hbar,q,z)$.

### 1.3.

From (1.2) it is clear that the vectors

(1.10)
$\displaystyle{\Psi}_{i}({\boldsymbol{u}},\hbar,q,z)=\left(\begin{array}[]{l}\tilde{F}_{i}({\boldsymbol{u}},\hbar,q,z)\\\
\tilde{F}_{i}({\boldsymbol{u}},\hbar,q,zq)\\\ \dots\\\
\tilde{F}_{i}({\boldsymbol{u}},\hbar,q,zq^{n-1})\end{array}\right)$

satisfy a first order $q$-difference equation

(1.11)
$\displaystyle{\Psi}_{i}({\boldsymbol{u}},\hbar,q,zq)=\mathbf{M}_{\mathcal{L}}({\boldsymbol{u}},\hbar,q,z)\,{\Psi}_{i}({\boldsymbol{u}},\hbar,q,z)$

for all $i=1,\dots,n$, where
$\mathbf{M}_{\mathcal{L}}({\boldsymbol{u}},\hbar,q,z)$ is the companion matrix
for the characteristic polynomial of (1.3):

$\mathbf{M}_{\mathcal{L}}({\boldsymbol{u}},\hbar,q,z)=\left(\begin{array}[]{cccccc}0&1&0&0&\dots&0\\\
0&0&1&0&\dots&0\\\ 0&0&0&1&\dots&0\\\ ..&..&..&..&..&..\\\
\frac{\bar{\alpha}_{n}(z-1)}{1-\hbar^{n}z}&\frac{\bar{\alpha}_{n-1}(z\hbar-1)}{1-\hbar^{n}z}&\frac{\bar{\alpha}_{n-2}(z\hbar^{2}-1)}{\hbar^{n}z-1}&\frac{\bar{\alpha}_{n-3}(z\hbar^{3}-1)}{1-\hbar^{n}z}&\dots&\frac{\bar{\alpha}_{1}(z\hbar^{n-1}-1)}{1-\hbar^{n}z}\\\
\end{array}\right)$

where $\bar{\alpha}_{k}$ denotes the coefficients of $x^{k}$ in
$\prod_{i=1}^{n}(1-xu_{i})$. The matrix

${\Psi}({\boldsymbol{u}},\hbar,q,z)=\Big{(}{\Psi}_{1}({\boldsymbol{u}},\hbar,q,z),\dots,{\Psi}_{n}({\boldsymbol{u}},\hbar,q,z)\Big{)}.$

with $i$-th column given by ${\Psi}_{i}({\boldsymbol{u}},\hbar,q,z)$ is a
fundamental solution matrix of the $q$-hypergeometric system
$\mathcal{H}({\boldsymbol{u}},\hbar,q,z)$. Explicitly, it has the following
matrix elements

(1.12)
$\displaystyle{\Psi}_{i,j}({\boldsymbol{u}},\hbar,q,z)=\tilde{F}_{j}({\boldsymbol{u}},\hbar,q,zq^{i-1}).$

Taking a matrix with columns given by $\mathbb{K}$ \- linear combinations of
the columns of $\tilde{\Psi}({\boldsymbol{u}},\hbar,q,z)$ produces new
fundamental solution matrix. Thus other fundamental solution matrices are of
the form

${\Psi}({\boldsymbol{u}},\hbar,q,z)\Lambda,\ \
\Lambda\in\mathsf{GL}_{n}(\mathsf{\mathbb{K}}).$

Finally, we also note that the fundamental solution matrix has the form

(1.13)
$\displaystyle{\Psi}({\boldsymbol{u}},\hbar,q,z)={\Psi}^{\prime}({\boldsymbol{u}},\hbar,q,z)\textsf{E}(z).$

where $\textsf{E}(z)$ is the diagonal matrix
$\textsf{E}(z)=\operatorname{diag}(e_{1}(z),e_{2}(z),\dots,e_{n}(z))$, and
${\Psi}^{\prime}({\boldsymbol{u}},\hbar,q,z)$ is analytic near $z=0$, i.e., is
a power series in $z$.

### 1.4.

Throughout this paper we will be using both, the multiplicative parameters
${\boldsymbol{u}}=(u_{1},\dots,u_{n})$, $\hbar$ and additive parameters
$\boldsymbol{a}=(a_{1},\dots,a_{n})$, $h$ which are related by

(1.14) $\displaystyle u_{1}=q^{a_{1}},\dots,u_{n}=q^{a_{n}},\ \ \hbar=q^{h}.$

##  2\. Vertex of $X=T^{*}\mathbb{P}^{n-1}$

The $q$-hypergeometric system $\mathcal{H}({\boldsymbol{u}},\hbar,q,z)$
describes equivariant quantum $K$-theory of the cotangent bundles over
projectile spaces. This set up will be the most convenient to us and we
briefly describe it in this section.

### 2.1.

Let $(\mathbb{C}^{\times})^{n}$ be the torus acting on $\mathbb{C}^{n}$ by
scaling the coordinate lines with characters $u_{1},\dots,u_{n}$. We have an
induced action of this torus on the cotangent bundle over projectile space
$X=T^{*}\mathbb{P}(\mathbb{C}^{n})$. Let
$T=(\mathbb{C}^{\times})^{n}\times\mathbb{C}^{\times}$ be a bigger torus
acting on $X$: the first factor $(\mathbb{C}^{\times})^{n}$ acts as before and
the second $\mathbb{C}^{\times}$ scales the cotangent directions with a
character $\hbar^{-1}$. Recall that the $T$-equivariant K-theory of $X$ is a
ring:

(2.1) $\displaystyle
K_{T}(X)=\mathbb{Z}[L^{\pm},u_{1}^{\pm},\dots,u^{\pm}_{n},\hbar^{\pm}]/(L-u_{1})\dots(L-u_{n}).$

where $L$ is the class of the line bundle $\mathcal{O}(1)$ and abusing
notations we denote by the same symbols $u_{1},\dots,u_{n}$ the trivial
equivariant line bundles associated to the characters $u_{i}$.

In this language, the additive parameters $a_{i}$, $h$ may be thought of as
the first Chern classes of the line bundles $u_{i}$ and $\hbar$. In the same
way, let $x$ be the first Chern class of $L$, i.e., we define $L=q^{x}$.

### 2.2.

Let us define a formal extension of this ring

$\hat{K}_{T}(X)=\mathrm{completion\ of}\ \ K_{T}(X)[z^{x}][[q]],$

so that it contains the following gamma class:

(2.2)
$\displaystyle\Phi(\boldsymbol{a},h,x,q,z)=\Big{(}\dfrac{(q^{h},q)_{\infty}}{(q,q)_{\infty}}\Big{)}^{\\!\\!n}\,\prod\limits_{i=1}^{n}\dfrac{(q^{\,x+1-a_{i}},q)_{\infty}}{(q^{\,x+h-a_{i}},q)_{\infty}}\,z^{\,x}\in\hat{K}_{T}(X),$

where $(u,q)_{\infty}=\prod\limits_{m=0}^{\infty}\,(1-uq^{m})$ is the
reciprocal of the $q$ \- gamma function. The normalized vertex function of $X$
is a generating functions counting equivariant quasimaps to quiver varieties,
see Section 7.2 of [Oko17] for definitions. For $X$ this functions has the
following form:

(2.3)
$\displaystyle\widetilde{V}(\boldsymbol{a},x,h,q,z)=\sum\limits_{d=0}^{\infty}\,\Phi(\boldsymbol{a},h,x+d,q,z)\,\in\hat{K}_{T}(X)[[z]]$

One verifies that

$P({\boldsymbol{u}},\hbar,z)\,\widetilde{V}(\boldsymbol{a},x,h,q,z)=(L-u_{1})(L-u_{2})\dots(L-u_{n})\,\Phi(\boldsymbol{a},h,x,q,z)$

where $P({\boldsymbol{u}},\hbar,z)$ is the $q$-difference operator given by
(1.3). This means that in (2.1) the vertex function solves the
$q$-hypergeometric equation (1.2).

### 2.3.

The components of the class (2.3) in any chosen basis of $K$-theory (2.1)
provide a basis of solutions of the $q$-hypergeometric system
$\mathcal{H}({\boldsymbol{u}},\hbar,q,z)$. For example, one convenient basis
consists of the $K$-theory classes of the $T$-fixed points $X^{T}$. The
components of the class (2.3) in this basis are obtained by the substitution
$L=u_{i}$ or, in the additive notations, by the substitution $x=a_{i}$. In
this way we obtain the basis of solutions given, up to a normalization
multiple, by the hypergeometric series (1.5):

(2.4)
$\displaystyle\widetilde{V}(\boldsymbol{a},a_{i},h,q,z)=\Phi(\boldsymbol{a},h,a_{i},q,z)\tilde{F}_{i}(\boldsymbol{a},h,q,z).$

Therefore, the vertex function (2.3) provides a uniform “basis free”
description of solutions for the $q$-hypergeometric system
$\mathcal{H}({\boldsymbol{u}},\hbar,q,z)$.

### 2.4.

It will be convenient to normalize all solutions as in the vertex (2.4). In
particular, we introduce the normalized fundamental solution matrix

$\tilde{\Psi}(\boldsymbol{a},h,q,z)=\Big{(}\tilde{\Psi}_{1}(\boldsymbol{a},h,q,z),\dots,\tilde{\Psi}_{n}(\boldsymbol{a},h,q,z)\Big{)},$

where

$\tilde{\Psi}_{i}(\boldsymbol{a},h,q,z)={\Psi}_{i}(\boldsymbol{a},h,q,z)\Phi(\boldsymbol{a},h,a_{i},q,z)$

and ${\Psi}_{i}(\boldsymbol{a},h,q,z)$ are given by (1.10). We can also
rewrite the last formula simply as

$\tilde{\Psi}(\boldsymbol{a},h,q,z)={\Psi}(\boldsymbol{a},h,q,z)\Phi(\boldsymbol{a},h,x,q,z)$

where $\Phi(\boldsymbol{a},h,x,q,z)$ is now understood as the operator of
multiplication by the gamma class (2.2) in the equivariant K - theory. In
particular, in the basis of the $T$-fixed points this operator is represented
by the diagonal matrix with eigenvalues the
$\Phi(\boldsymbol{a},h,a_{i},q,z)$, $i=1,\dots,n$.

### 2.5.

The normalized fundamental solution matrices
$\tilde{\Psi}(\boldsymbol{a},h,q,z)$ also have an enumerative meaning: these
are the matrices of the capping operator for $X$, which is the partition
function of the relative quasimaps to $X$, see Section 7.4 in [Oko17] for the
definitions.

An important input from the enumerative geometry we need for our goals is the
existence of the shift operators. The operators describe $q$-difference
equations satisfied by the capping operator in the equivariant parameters
$\boldsymbol{a}$ and $h$. These equations exist in general for an arbitrary
Nakajima variety. Applied to $X=T^{*}\mathbb{P}^{n-1}$ this results can be
formulated as follows:

######  Theorem 2.1 (Theorem 8.2.20, [Oko17]).

The normalized fundamental solution matrices satisfy the following system of
difference equations

$\displaystyle\tilde{\Psi}(a_{1},\dots,a_{i}+1,\dots
a_{n},h,q,z)=S^{a_{i}}(\boldsymbol{a},h,q,z)\tilde{\Psi}(a_{1},\dots,a_{i},\dots
a_{n},h,q,z),$ $\displaystyle\tilde{\Psi}(a_{1},\dots,a_{i},\dots
a_{n},h+1,q,z)=S^{h}(\boldsymbol{a},h,q,z)\tilde{\Psi}(a_{1},\dots,a_{i},\dots
a_{n},h,q,z).$

where $S^{a_{i}}(\boldsymbol{a},h,q,z)$ and $S^{h}(\boldsymbol{a},h,q,z)$
denote matrices whose entries are rational functions in the equivariant
parameters ${\boldsymbol{u}}$, $\hbar$, $q$ and $z$:

$S^{a_{i}}(\boldsymbol{a},h,q,z),S^{h}(\boldsymbol{a},h,q,z)\in
GL_{n}(\mathbb{Q}({\boldsymbol{u}},\hbar,z,q)).$

##  3\. $p$ -adic $q$-hypergeometric functions

### 3.1.

Let $p$ be a prime, $\mathbb{Q}_{p}$ be the filed of $p$-adic numbers,
$\mathbb{Z}_{p}$ be the integers of $\mathbb{Q}_{p}$. Starting from this
section $\mathbb{K}$ denotes the $p$-adic completion of the algebraic closure
of $\mathbb{Q}_{p}$. Let us fix $q\in\mathbb{K}^{\times}$ of the following
form

(3.1) $\displaystyle q=1+t,\ \ t\neq 0,\ \ |t|_{p}\leqslant 1.$

From now, we assume that all the equivariant parameters (1.14) are $p$ \- adic
integers $a_{1},\dots,a_{n},h\in\mathbb{Z}_{p}$.

### 3.2.

Let $\overline{\mathbb{K}(z)}$ be the completion of $\mathbb{K}(z)$ with
respect to the Gauss norm:

$\left|\sum\limits_{m=0}^{\infty}\,r_{m}(z)\right|_{\mathrm{Gauss}}=\ \
\max\,\\{\,|r_{m}(z)|_{p}:m=0,1,2,\dots\\}.$

A Frobenius intertwiner from $\mathcal{H}(\boldsymbol{a},h,q^{p},z^{p})$ to
$\mathcal{H}(\boldsymbol{a}^{\prime},h^{\prime},q,z)$ is
$\mathsf{U}(z)\in\mathsf{GL}_{n}(\overline{\mathbb{K}(z)})$ such that

$\mathbf{M}_{\mathcal{L}}(\boldsymbol{a}^{\prime},\hbar^{\prime},q,z)=\mathsf{U}(zq)^{-1}\,\mathbf{M}_{\mathcal{L}}(\boldsymbol{a},\hbar,q^{p},z^{p})\,\mathsf{U}(z).$

If ${\Psi}(\boldsymbol{a},\hbar,q^{p},z^{p})$ is a fundamental solution matrix
for the hypergeometric system $\mathcal{H}(\boldsymbol{a},\hbar,q^{p},z^{p})$
(1.11) then $\mathsf{U}(z){\Psi}(\boldsymbol{a},\hbar,q^{p},z^{p})$ is a
fundamental solution matrix for
$\mathcal{H}(\boldsymbol{a}^{\prime},h^{\prime},q,z)$. Thus, it is of the form
${\Psi}(\boldsymbol{a}^{\prime},\hbar^{\prime},q,z)\Lambda$ and we obtain a
relation

(3.2)
$\displaystyle{\Psi}(\boldsymbol{a}^{\prime},\hbar^{\prime},q,z)\Lambda=\mathsf{U}(z){\Psi}(\boldsymbol{a},\hbar,q^{p},z^{p})$

for some constant matrix $\Lambda\in\mathsf{GL}_{n}(\mathsf{\mathbb{K}})$. The
matrix $\mathsf{U}(0)$ is called the constant term of the Frobenius
intertwiner. If a fundamental solution is normalized by
${\Psi}(\boldsymbol{a},\hbar,q,0)=1$, then $\mathsf{U}(0)=\Lambda$.

### 3.3.

For the purposes of this note, it is sufficient to restrict to a smaller ring
$\widehat{\mathsf{R}}_{p}\subset\overline{\mathbb{K}(z)}$ which has a more
explicit structure. Let $\mathsf{R}_{p}=\mathbb{Z}[z,(1-z)^{-1}]$ be the ring
of rational functions with denominator given by powers of $(1-z)^{-1}$. The
ring we need is the $p$-adic completion

$\widehat{\mathsf{R}}_{p}=\lim\limits_{\leftarrow}\,\mathsf{R}_{p}/p^{s}\mathsf{R}_{p}.$

By definition, the elements of $\widehat{\mathsf{R}}_{p}$ are sums of the form

$\sum\limits_{k=0}^{\infty}\,p^{k}\dfrac{g_{k}(z)}{(1-z)^{m_{k}}}$

where $m_{k}\in\mathbb{N}$ and $g_{k}(z)\in\mathbb{Z}[z]$. Put differently, an
element $g(z)\in\widehat{\mathsf{R}}_{p}$ can be represented by a power series
$g(z)\in\mathbb{Z}[[z]]$ such that for any $s\geqslant 0$ there is a natural
number $m_{s}$ such that

$(1-z)^{m_{s}}g(z)\\!\\!\pmod{p^{s+1}}\in\mathbb{Z}[z].$

This condition provides a useful numerical test to determine whether a given
power series $g(z)$ belongs to $\widehat{\mathsf{R}}_{p}$ or not, see Section
5. The Frobenius intertwiners we encounter in this paper are all
$\mathsf{GL}_{n}(\widehat{\mathsf{R}}_{p})$ \- valued.

### 3.4.

For $n\in\mathbb{N}$ we define the $q$-number by:

$[n]_{q}=\dfrac{1-q^{n}}{1-q}=1+q+\dots+q^{n-1}.$

Let $[n]_{q}!=[1]_{q}[2]_{q}\dots[n]_{q}$ denote the $q$-factorial and

(3.3)
$\displaystyle\binom{n}{k}_{\\!\\!q}=\dfrac{[n]_{q}!}{[k]_{q}![n-k]_{q}!}$

be the $q$-binomial factor. We recall the original definitions of the $p$-adic
gamma and $q$-gamma functions:

######  Theorem 3.1 (Koblitz, [Ko80]).

For $q$ as in (3.1) the function
$\Gamma_{p,q}(n)=(-1)^{n}\prod\limits_{{0<i<n}\atop{p\nmid i}}\,[n]_{q}$
extends to a continuous function on $\mathbb{Z}_{p}$. $\lim\limits_{q\to
1}\,\Gamma_{p,q}=\Gamma_{p}$ where $\Gamma_{p}$ denotes the $p$-adic gamma
function of Morita [Mo75], which is the extension to $\mathbb{Z}_{p}$ of
$\Gamma_{p}(n)=(-1)^{n}\prod\limits_{{0<i<n}\atop{p\nmid i}}\,n.$

### 3.5.

We formulate our main theorem in the geometric language of Section 2. For
this, we replace the equivariant K-theory (2.1) by

(3.4) $\displaystyle
K_{T}(X,\mathbb{K})=K_{T}(X)\otimes\mathbb{K}=\mathbb{K}[L^{\pm},u_{1}^{\pm},\dots,u^{\pm}_{n},\hbar^{\pm}]/(L-u_{1})\dots(L-u_{n}).$

The Frobenius intertwiner is understood as a map

$\mathsf{U}(z):K_{T}(X,\mathbb{K})[[z]]\longrightarrow
K_{T}(X,\mathbb{K})[[z]],$

In particular, the constant term acts on $K$-theory

$\mathsf{U}(0):K_{T}(X,\mathbb{K})\longrightarrow K_{T}(X,\mathbb{K}).$

We are now ready to formulate our main result:

######  Theorem 3.2.

For any $\boldsymbol{a}=(a_{1},\dots,a_{n})\in\mathbb{Z}^{n}_{p},\ \
h\in\mathbb{Z}_{p}$ there exists unique Frobenius intertwiner
$\mathsf{U}(z)\in\mathsf{GL}_{n}(\overline{\mathbb{K}(z)})$ from
$\mathcal{H}(\boldsymbol{a},h,q^{p},z^{p})$ to
$\mathcal{H}(p\boldsymbol{a},ph,q,z)$, where
$p\boldsymbol{a}=(pa_{1},\dots,pa_{n})$. The constant term of the intertwiner
acts in $K_{T}(X,\mathbb{K})$ as the operator of multiplication by the
K-theory class

(3.5)
$\displaystyle\mathsf{U}(0)=\prod\limits_{j=1}^{n}\,\dfrac{\Gamma_{p,q}(px+ph-
pa_{j})}{\Gamma_{p,q}(px-pa_{j})\Gamma_{p,q}(ph)}.$

Clearly, multiplying a Frobenius intertwiner by a constant multiple
$\mathsf{U}(z)\to c\mathsf{U}(z)$, $c\in\mathbb{K}^{\times}$ preserves all the
properties. So, the “uniqueness” in Theorem 3.2 should be understood as
uniqueness up to a scalar multiple.

If the torus fixed points of $T$-action are isolated, multiplication by any
equivariant K-theory class is diagonal in the basis given by the classes of
$T$-fixed points. In our case $X=T^{*}\mathbb{P}^{n-1}$ the $X^{T}$ is a
finite set when all $a_{1},\dots,a_{n}$ are pairwise distinct. The
corresponding fixed point component of $\mathsf{U}(0)$ are obtained by
specialization $x=a_{i}$. We thus conclude with the following result.

######  Corollary 3.3.

If all $a_{1},\dots,a_{n}$ are pairwise distinct, the constant term of the
Frobenius intertwiner $\mathsf{U}(0)$ acts in the basis of torus fixed points
as a diagonal matrix with the following eigenvalues:

(3.6)
$\displaystyle\mathsf{U}(0)_{i,i}=\prod\limits_{j=1}^{n}\,\dfrac{\Gamma_{p,q}(p(a_{i}-a_{j}+h))}{\Gamma_{p,q}(p(a_{i}-a_{j}))\Gamma_{p,q}(ph)}.$

######  Remark.

In the limit $q\to 1$ when $\Gamma_{p,q}\to\Gamma_{p}$ and Theorem 3.2
specializes to the results of Dwork [Dw89] for hypergeometric equations, see
also [Ked21] for recent treatment. We consider this case in more details in
Section 6.

##  4\. Proof of Theorem 3.2

### 4.1.

Let $t\in\mathbb{Q}_{p}$ with $|t|_{p}\leqslant|p|_{p}=1/p$. Let $q=1+t$.
Using elementary combinatorics one can derive the following congruence

(4.1) $\displaystyle(z,q)_{p^{s}}\equiv(z^{p},q^{p})_{p^{s-1}}\pmod{p^{s}}$

for instance, using the induction on $s$. Let us introduce the symbol

$[x,q]_{d}=(1-qx)\dots(1-q^{d-1}x)=\dfrac{(xq,q)_{\infty}}{(xq^{d},q)_{\infty}}$

which differs from the standard Pochhammer by a factor.

######  Lemma 4.1.

For any $i\in\mathbb{Z}$, modulo $p^{s}$ we have

$\dfrac{[q^{i},q]_{p^{s}}}{[1,q]_{p^{s}}}\equiv\left\\{\begin{array}[]{ll}\dfrac{[q^{i},q^{p}]_{p^{s-1}}}{[1,q^{p}]_{p^{s-1}}},&\mathrm{if}\
\ p\mid i,\\\ &\\\ 0,&\mathrm{if}\ \ p\nmid i.\end{array}\right.$

###### Proof.

By the $q$-binomial theorem

$\sum\limits_{i=0}^{\infty}\,\dfrac{[q^{i},q]_{p^{s}}}{[1,q]_{p^{s}}}z^{i}=\dfrac{1}{(z,q)_{p^{s}}},\
\ \
\sum\limits_{i=0}^{\infty}\,\dfrac{[q^{pi},q^{p}]_{p^{s-1}}}{[1,q^{p}]_{p^{s-1}}}z^{pi}=\dfrac{1}{(z^{p},q^{p})_{p^{s-1}}},$

thus, for $i\geqslant$ the result follows from (4.1). Next for
$-p^{s}+1\leqslant i\leqslant-1$

$[q^{i},q]_{p^{s}}=0$

and similarly for $[q^{i},q^{p}]_{p^{s-1}}$, thus, Lemma holds for these $i$’s
too. Finally, if $i\leqslant-p^{s}$, the Lemma follows from the identity

$[q^{i},q]_{p^{s}}={q^{(p^{s}-1)i}q^{\frac{p^{s}(p^{s}-1)}{2}}}\,[q^{-(i+p^{s})},q]_{p^{s}}.$

∎

### 4.2.

Let us consider the normalized vertex function (2.3) at $h=p^{s}$. We find

$\widetilde{V}(\boldsymbol{a},x,p^{s},q,z)=z^{x}\,\sum\limits_{d=0}^{\infty}\,\Big{(}\prod\limits_{i=1}^{n}\dfrac{[q^{\,x+d-a_{i}},q]_{p^{s}}}{[1,q]_{p^{s}}}\Big{)}z^{d}.$

######  Theorem 4.2.

We have the following congruences for the normalized K-theoretic vertex
functions:

$\tilde{V}(p\boldsymbol{a},px,p^{s},q,z)\equiv\tilde{V}(\boldsymbol{a},x,p^{s-1},q^{p},z^{p})\pmod{p^{s}}.$

###### Proof.

The normalized K-theoretic vertex function is

$\tilde{V}(p\boldsymbol{a},px,p^{s},q,z)=z^{px}\,\sum\limits_{d=0}^{\infty}\,\left(\prod\limits_{i=1}^{n}\dfrac{[q^{px-
pa_{i}+d},q]_{p^{s}}}{[1,q]_{p^{s}}}\right)z^{d}$

Lemma 4.1 gives

$z^{px}\,\sum\limits_{d=0}^{\infty}\,\left(\prod\limits_{i=1}^{n}\dfrac{[q^{px-
pa_{i}+d},q]_{p^{s}}}{[1,q]_{p^{s}}}\right)z^{d}\equiv
z^{px}\,\sum\limits_{d=0}^{\infty}\,\left(\prod\limits_{i=1}^{n}\dfrac{[q^{px-
pa_{i}+pd},q^{p}]_{p^{s-1}}}{[1,q^{p}]_{p^{s-1}}}\right)z^{pd}\pmod{p^{s}}$

which is exactly the statement of the theorem. ∎

Since the vertex describes all solutions of the $q$-holonomic system we also
obtain:

######  Corollary 4.3.

The normalized fundamental solution matrices satisfy the following
congruences:

$\widetilde{\Psi}(p\boldsymbol{a},p^{s},q,z)\pmod{p^{s}}=\widetilde{\Psi}(\boldsymbol{a},p^{s-1},q^{p},z^{p}).$

### 4.3.

To prove Theorem 3.2 it is enough to show that for any $s\in\mathbb{N}$,

$\mathsf{U}(z)=\widetilde{\Psi}(p\boldsymbol{a},ph,q,z)\,\widetilde{\Psi}(\boldsymbol{a},h,q^{p},z^{p})^{-1}\pmod{p^{s}},$

is a rational function in $z$. Let us write $h=h_{0}+p^{s-1}h_{1}$ for some
$h_{0}\in\mathbb{N}$ and $h_{i}\in\mathbb{Z}_{p}$. By Theorem 2.1 we have:

$\mathsf{U}(z)=\widetilde{\Psi}(p\boldsymbol{a},ph,q,z)=\widetilde{\Psi}(p\boldsymbol{a},ph_{0}+h_{1}p^{s},q,z)=\left(\prod\limits_{i=h_{1}p^{s}}^{ph_{0}+h_{1}p^{s}-1}S^{h}(p\boldsymbol{a},i,q,z)\right)\widetilde{\Psi}(p\boldsymbol{a},h_{1}p^{s},q,z),$

where $S^{h}(p\boldsymbol{a},i,q,z)$ is a certain invertible matrix whose
coefficients are rational functions of $z$. By Corollary 4.3 we obtain:

$\widetilde{\Psi}(p\boldsymbol{a},ph,q,z)\equiv\left(\prod\limits_{i=0}^{ph_{0}-1}S^{h}(p\boldsymbol{a},i,q,z)\right)\widetilde{\Psi}(\boldsymbol{a},h_{1}p^{s-1},q^{p},z^{p})\pmod{p^{s}}.$

Similarly, we obtain

$\widetilde{\Psi}(\boldsymbol{a},h,q^{p},z^{p})\equiv\left(\prod\limits_{i=0}^{h_{0}-1}S^{h}(\boldsymbol{a},i,q^{p},z^{p})\right)\widetilde{\Psi}(\boldsymbol{a},h_{1}p^{s-1},q^{p},z^{p})$

and thus

(4.2)
$\displaystyle\widetilde{\Psi}(p\boldsymbol{a},ph,q,z)\,\widetilde{\Psi}(\boldsymbol{a},h,q^{p},z^{p})^{-1}\equiv\left(\prod\limits_{i=0}^{ph_{0}-1}S^{h}(p\boldsymbol{a},i,q,z)\right)\left(\prod\limits_{i=0}^{h_{0}-1}S^{h}(\boldsymbol{a},i,q^{p},z^{p})\right)^{-1}\pmod{p^{s}}$

which is a product of finitely many matrices whose entries are rational
functions in $z$.

### 4.4.

Finally, from the normalization (2.2), (2.3) we find:

$\mathsf{U}(0)=\dfrac{\Phi(p\boldsymbol{a},ph,px,q,z)}{\Phi(\boldsymbol{a},h,x,q^{p},z^{p})}$

From definition (2.2) we see that $\mathsf{U}(0)$ is exactly the product of
factors as in Lemma 4.4 below, and therefore

$\mathsf{U}(0)=\prod\limits_{j=1}^{n}\,\dfrac{\Gamma_{p,q}(px+ph-
pa_{j})}{\Gamma_{p,q}(px-pa_{j})\Gamma_{p,q}(ph)}$

which finishes the proof of the theorem.

### 4.5.

If $n$ and $h$ are two natural numbers then

$\dfrac{(q^{ph},q)_{\infty}}{(q,q)_{\infty}}\dfrac{(q^{pn+1},q)_{\infty}}{(q^{pn+ph},q)_{\infty}}=\binom{pn+ph-1}{pn}_{\\!\\!q}.$

is a $q$-binomial coefficient (3.3). Similarly

$\dfrac{(q^{ph},q^{p})_{\infty}}{(q^{p},q^{p})_{\infty}}\dfrac{(q^{pn+p},q^{p})_{\infty}}{(q^{pn+ph},q^{p})_{\infty}}=\binom{n+h-1}{n}_{\\!\\!q^{p}}.$

For the ratio of these factors we find:

$\dfrac{\binom{pn+ph-1}{pn}_{\\!\\!q}}{\binom{n+h-1}{n}_{\\!\\!q^{p}}}=\dfrac{\prod\limits_{{0<i<pn+ph}\atop{p\nmid
i}}\,[i]_{q}}{(\prod\limits_{{0<i<pn}\atop{p\nmid
i}}\,[i]_{q})(\prod\limits_{{0<i<ph}\atop{p\nmid
i}}\,[i]_{q})}=\dfrac{\Gamma_{p,q}(pn+ph)}{\Gamma_{p,q}(pn)\,\Gamma_{p,q}(ph)}$

where the last equality is by definition of $(p,q)$-Gamma function. By
Koblitz’s Theorem 3.1 we obtain

######  Lemma 4.4.

The function of $n,h\in\mathbb{N}$ given by

$\dfrac{\dfrac{(q^{ph},q)_{\infty}}{(q,q)_{\infty}}\dfrac{(q^{pn+1},q)_{\infty}}{(q^{pn+ph},q)_{\infty}}}{\dfrac{(q^{ph},q^{p})_{\infty}}{(q^{p},q^{p})_{\infty}}\dfrac{(q^{pn+p},q^{p})_{\infty}}{(q^{pn+ph},q^{p})_{\infty}}}=\dfrac{\Gamma_{p,q}(pn+ph)}{\Gamma_{p,q}(pn)\,\Gamma_{p,q}(ph)}$

extends to $\mathbb{Z}_{p}$ values of arguments $n$ and $h$.

##  5\. Numerical examples of Theorem 3.2

### 5.1.

Let ${\Psi}(\boldsymbol{a},\hbar,q,z)$ be the fundamental solution matrix
(1.12) for some fixed parameters. As in (3.2) we look for the Frobenius
intertwiner matrix in the form

(5.1)
$\displaystyle\mathsf{U}(z)={\Psi}(p\boldsymbol{a},ph,q,z)\Lambda{\Psi}(\boldsymbol{a},h,q^{p},z^{p})^{-1}.$

where $\Lambda$ is a constant matrix. In order to be a Frobenius intertwiner,
we need to find such $\Lambda\in\mathsf{GL}_{n}(\mathbb{K})$ for which (5.1)
is an element of $\mathsf{GL}_{n}(\widehat{\mathsf{R}}_{p})$. Explicitly,
$\mathsf{U}(z)$ must be a power series

$\mathsf{U}(z)=\mathsf{U}_{0}+\mathsf{U}_{1}z+\dots\in\mathsf{Mat}_{n}(\mathbb{Z})[[z]]$

such that for any natural number $s$, there exists a natural number $m_{s}$
for which

(5.2)
$\displaystyle(1-z)^{m_{s}}\,\mathsf{U}(z)\pmod{p^{s}}\in\mathsf{Mat}_{n}(\mathbb{Z})[z].$

In words: after multiplication by sufficiently large power of $(1-z)$ the
matrix elements of $\mathsf{U}(z)$ are polynomials in $z$ modulo $p^{s}$. This
condition is very strong and can be used to effectively compute the Frobenius
intertwiners as we about to show.

### 5.2.

Note that if the parameters $\boldsymbol{a}=(a_{1},\dots,a_{n})$ are pairwise
distinct the matrix $\Lambda$ must be diagonal. Indeed, the fundamental
solution matrix $\tilde{\Psi}(\boldsymbol{a},\hbar,q,z)$ contains logarithmic
terms given by matrix $\mathsf{E}(z)$ in (1.13). The intertwiner (5.1) is a
power series in $z$ only if these log terms are cancelled. It is elementary to
check that for the pairwise distinct $a_{i}$ this happens only if $\Lambda$ is
diagonal. Since $\Lambda$ is defined up to a scalar factor, we may assume that
$\Lambda_{n,n}=1$ so

(5.8) $\displaystyle\Lambda=\left(\begin{array}[]{ccccc}c_{1}&0&0&..&0\\\
0&c_{2}&0&..&0\\\ ..&..&..&..&..\\\ 0&0&..&c_{n-1}&0\\\
0&0&0&..&1\end{array}\right)$

for some unknown $c_{1},\dots,c_{n-1}\in\mathbb{Z}_{p}$.

### 5.3.

Let us assume $n=2$. In this case

(5.11) $\displaystyle\Lambda=\left(\begin{array}[]{cc}c&0\\\
0&1\end{array}\right).$

Let us pick a prime $p=3$. To meet the condition (3.1) let us also set

$q=1+p=4,$

and pick some random values of $a_{1},a_{2},h\in\mathbb{Z}_{p}$, for instance

(5.12) $\displaystyle h=\dfrac{1}{2},\ \ a_{1}=\dfrac{1}{5},\ \ a_{2}=0.$

(Since the solutions only depend on the difference $a_{1}-a_{2}$ we can always
assume $a_{2}=0$). For our computer check we use integral approximations of
these numbers say up to $p^{7}$:

$\bar{h}=h\pmod{p^{7}}=1094,\ \ \bar{a_{1}}=a_{1}\pmod{p^{7}}=875,\ \ \
\bar{a_{2}}=0.$

We now determine the constant $c\in\mathbb{Z}_{p}$ in (5.11) from the
rationality condition (5.2) in a chain of steps. For this we assume that $c$
has the following $p$-adic expansion:

$c=c^{(0)}+c^{(1)}3+c^{(2)}3^{2}+\dots$

where $c^{(i)}\in\\{0,1,2\\}$.

Step 1: determination of $c^{(0)}$ and $c^{(1)}.$

Using a computer we approximate the power series
${\Psi}(\boldsymbol{a},\hbar,q,z)$ by expansion up to $O(z^{20})$.
Substituting this approximation to (5.1) and trying the integers $0,1,2$ for
the values of $c^{(0)}$ and $c^{(1)}$ we find that

$\mathsf{U}(z)\pmod{p}$

exists (i.e. no powers of $p$ in the denominators) only if $c^{(0)}=1$ and
$c^{(1)}=0$. In which case we find that

(5.15)
$\displaystyle\mathsf{U}(z)\pmod{p}=\left(\begin{array}[]{cc}1+O\left({z}^{20}\right)&O\left({z}^{20}\right)\\\
\par\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
O\left({z}^{20}\right)&1+O\left({z}^{20}\right)\end{array}\right)$

We note that $\mathsf{U}(z)\pmod{p}$ given by (5.15) does not depend on the
value of $c^{(2)}$ and higher.

Step 2: determination of $c^{(2)}.$

If $c^{(2)}=0$, using a computer we find

$\mathsf{U}(z)\pmod{p}=\left[\begin{array}[]{cc}1+O\left({z}^{20}\right)&O\left({z}^{20}\right)\\\
\par\vskip 6.0pt plus 2.0pt minus
2.0pt\cr{z}^{12}+2\,{z}^{15}+O\left({z}^{20}\right)&1+O\left({z}^{20}\right)\end{array}\right]$

If $c^{(2)}=1$ we find

$\mathsf{U}(z)\pmod{p}=\left[\begin{array}[]{cc}1+O\left({z}^{20}\right)&O\left({z}^{20}\right)\\\
\par\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
O\left({z}^{20}\right)&1+O\left({z}^{20}\right)\end{array}\right]$

If $c^{(2)}=2$ we find

$\mathsf{U}(z)\pmod{p}=\left[\begin{array}[]{cc}1+O\left({z}^{20}\right)&O\left({z}^{20}\right)\\\
\par\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
2\,{z}^{12}+{z}^{15}+O\left({z}^{20}\right)&1+O\left({z}^{20}\right)\end{array}\right]$

We see that the only value of $c^{(2)}$ for which the result is a polynomial
in $z$ is $c^{(2)}=1$. These results do not depend on the value of the
coefficient $c^{(3)}$ and higher which are to be determined in the next step.

Step 3: determination of $c^{(3)}.$

If $c^{(3)}=0$ we find:

$(1-z)^{2}\mathsf{U}(z)\\!\\!\pmod{p^{2}}=\left[\begin{array}[]{cc}1+z+{z}^{2}+O\left({z}^{20}\right)&O\left({z}^{20}\right)\\\
\par\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
6\,z+3\,{z}^{12}+3\,{z}^{13}+3\,{z}^{14}+6\,{z}^{15}+6\,{z}^{16}+6\,{z}^{17}+O\left({z}^{20}\right)&1+7\,z+{z}^{2}+O\left({z}^{20}\right)\end{array}\right]$

If $c^{(3)}=1$ we find:

$(1-z)^{2}\mathsf{U}(z)\\!\\!\pmod{p^{2}}=\left[\begin{array}[]{cc}1+z+{z}^{2}+O\left({z}^{20}\right)&O\left({z}^{20}\right)\\\
\par\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
6\,z+O\left({z}^{20}\right)&1+7\,z+{z}^{2}+O\left({z}^{20}\right)\end{array}\right]$

If $c^{(3)}=2$ we find:

$(1-z)^{2}\mathsf{U}(z)\\!\\!\pmod{p^{2}}=\left[\begin{array}[]{cc}1+z+{z}^{2}+O\left({z}^{20}\right)&O\left({z}^{20}\right)\\\
\par\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
6\,z+6\,{z}^{12}+6\,{z}^{13}+6\,{z}^{14}+3\,{z}^{15}+3\,{z}^{16}+3\,{z}^{17}+O\left({z}^{20}\right)&1+7\,z+{z}^{2}+O\left({z}^{20}\right)\end{array}\right]$

We conclude that there is unique value of $c^{(3)}=1$ for which the result is
a polynomial in $z$.

Continuing this process we will find that at $m$-th step there is always
unique value of $c^{m}$ for which rationality condition (5.2) is met. For
instance, up to the order $9$ these method gives the following value

(5.16) $\displaystyle
c=1+3^{2}+3^{3}+2\,3^{4}+3^{5}+2\,3^{6}+3^{7}+3^{8}+O(3^{9}).$

### 5.4.

Theorem 3.2 and Corollary 3.3 gives the following formula for the constant
$c$:

$c=\dfrac{\Lambda_{1,1}}{\Lambda_{2,2}}=\dfrac{\Gamma_{p,q}(p(a_{1}-a_{2}+h))\Gamma_{p,q}(p(a_{2}-a_{1}))}{\Gamma_{p,q}(p(a_{2}-a_{1}+h))\Gamma_{p,q}(p(a_{2}-a_{1}))}$

at (5.12) we have

$c=\dfrac{\Gamma_{p,q}(\frac{21}{10})\Gamma_{p,q}(-\frac{3}{5})}{\Gamma_{p,q}(\frac{9}{10})\Gamma_{p,q}(\frac{3}{5})}.$

For $p=3$ and $q=1+p=4$ we compute:

$\dfrac{\Gamma_{p,q}(\frac{21}{10})\Gamma_{p,q}(-\frac{3}{5})}{\Gamma_{p,q}(\frac{9}{10})\Gamma_{p,q}(\frac{3}{5})}=1+3^{2}+3^{3}+2\,3^{4}+3^{5}+2\,3^{6}+3^{7}+3^{8}+O(3^{9})$

in full agreement with the value obtained from the numerical experiment
(5.16).

The author have performed the same numerical experiments for numerous other
rational values of parameters $a_{1},a_{2}$ and $h$ in $\mathbb{Z}_{p}$. In
all these cases the results are in agreement with Theorem 3.2. Same method can
be used for ranks $n>2$, in which case we have to determine the numerical
values of $n-1$ constants in (5.8). Again, we find full agreement with Theorem
3.2.

##  6\. Limit to cohomology

### 6.1.

As $q\to 1$, the quantum difference equation of $X$ degenerates to the quantum
differential equation, which governs quantum cohomology of $X$. In this limit,
the equivairnat K-theory ring (2.1) is replaced by the equivairnat cohomology

(6.1) $\displaystyle
H^{\bullet}_{T}(X)=\mathbb{Z}[x,a_{1},\dots,a_{n},h]/(x-a_{1})\dots(x-a_{n}).$

In this limit the gamma class (2.2) takes the form:

(6.2)
$\displaystyle\Gamma(\boldsymbol{a},h,x,z)=z^{x}\,\prod\limits_{i=1}^{n}\dfrac{\Gamma(x+h-a_{i})}{\Gamma(x+1-a_{i})\Gamma(h)}\in\hat{H}^{\bullet}_{T}(X).$

where $\Gamma$ denotes the classical Gamma function, and
$\hat{H}^{\bullet}_{T}(X)$ is the appropriate completion of the cohomology
ring containing the gamma class. The normalized cohomological vertex function
(also known as $J$-function in cohomology) is:

$\widetilde{V}(\boldsymbol{a},h,x,z)=\sum\limits_{d=0}^{\infty}\,\Gamma(\boldsymbol{a},h,x+d,z)\,\in\,\hat{H}^{\bullet}_{T}(X)[[z]].$

The vertex function satisfy the differential equation

(6.3) $\displaystyle
P(\boldsymbol{a},h,z)\widetilde{V}(\boldsymbol{a},h,x,z)=(x-a_{1})\dots(x-a_{n})\Gamma(\boldsymbol{a},h,x,z)=0$

(note that the right side is zero in (6.1)), where

(6.4) $\displaystyle
P(\boldsymbol{a},h,z)=z\prod\limits_{i=1}^{n}\,(D+h-a_{i})-\prod\limits_{i=1}^{n}(D-a_{i})$

and $D=z\frac{d}{dz}$. The components of the vertex in some basis of
$H^{\bullet}_{T}(X)$ provide a basis of solutions for the hypergeometric
differential equation (6.3). In particular, in the basis given by the classes
of the $T$-fixed points $X^{T}$, we obtain the standard generazlized
hypergeometric series:

$\widetilde{V}(\boldsymbol{a},h,a_{i},z)=\Gamma(\boldsymbol{a},h,a_{i},z)\,\sum\limits_{d=0}^{\infty}\,\left(\prod\limits_{j=1}^{n}\dfrac{(a_{i}+h-a_{j})_{d}}{(a_{i}+1-a_{j})_{d}}\right)z^{d}$

where $(a)_{d}=a(a+1)\dots(a+d-1)$.

By Theorem 3.1 $\Gamma_{p,q}(n)\stackrel{{\scriptstyle q\to
1}}{{\longrightarrow}}\Gamma_{p}(n)$ and from Theorem 3.2 we deduce:

######  Theorem 6.1.

For any $\boldsymbol{a}=(a_{1},\dots,a_{n})\in\mathbb{Z}^{n}_{p},\ \
h\in\mathbb{Z}_{p}$ there exists unique Frobenius intertwiner $\mathsf{U}(z)$
from the hypergeometric system (6.4) with parameters
$(a_{1},\dots,a_{n},h,z^{p})$ to the system with parameters
$(pa_{1},\dots,pa_{n},ph,z)$. The constant terms of the intertwiner is
represented by the operator acting in $\hat{H}^{\bullet}_{T}(X)$ as an
operator of multiplication by the cohomology class class

(6.5)
$\displaystyle\mathsf{U}(0)=\prod\limits_{j=1}^{n}\,\dfrac{\Gamma_{p}(px+ph-
pa_{j})}{\Gamma_{p}(px-pa_{j})\Gamma_{p}(ph)},$

where $\Gamma_{p}$ is the Morita gamma function.

The analog of Corollary 3.3 also gives:

######  Corollary 6.2.

If $a_{1},\dots,a_{n}$ are pairwise distinct, the constant term of the
Frobenius intertwiner in the basis of torus fixed points is given by diagonal
matrix with the following eigenvalues

(6.6)
$\displaystyle\mathsf{U}(0)_{i,i}=\prod\limits_{j=1}^{n}\,\dfrac{\Gamma_{p}(p(a_{i}-a_{j}+h))}{\Gamma_{p}(p(a_{i}-a_{j}))\Gamma_{p}(ph)}.$

In this way we arrive at the description of the Frobenius intertwines for the
$p$-adic hypergeometric equations discovered by Dwork [Dw89] see also [Ked21]
where this result was recently revisited by other tools.

######  Remark.

Recall that for $n\in p\mathbb{Z}_{p}$ the Morita Gamma function satisfies the
identities $\Gamma_{p}(n)=-\Gamma_{p}(1+n)=(-1)^{n_{0}}/\Gamma_{p}(-n)$. Using
this, up to a scalar multiple, the constant term of Frobenius (6.5) takes the
form

$\mathsf{U}(0)=\prod\limits_{i=1}^{n}\,\Gamma_{p}(px+ph-
pa_{j})\Gamma_{p}(pa_{j}-px)$

The arguments of gamma functions in this product are exactly the $p$-multiples
of the Chern roots of the tangent bundle $TX$. We thus can write this formally
as

(6.7) $\displaystyle\mathsf{U}(0)=\Gamma_{p}(TX^{(p)}),$

where $(p)$ indicates the Frobenius twist multiplying the Chern roots by $p$.
We expect that (6.7) holds for the constant term of the Frobenius intertwiner
of the quantum differential equation for an arbitrary $X$.

##  7\. Limit to $X=\mathbb{P}^{n-1}$: Frobenius structure for generalized
Bessel equations

### 7.1.

The quantum differential equation and its solutions for $X=\mathbb{P}^{n-1}$
can be obtained as the scaling limit $\hbar\to\infty$ of the corresponding
equations and solutions for $T^{*}\mathbb{P}^{n-1}$ (we recall that the
equivariant parameter $\hbar$ is the character corresponding to the dilation
of the cotangent directions of $T^{*}\mathbb{P}^{n-1}$ under the torus action)
[SV24]. In this section we consider the Frobenius intertwiners for the quantum
differential equation of $\mathbb{P}^{n-1}$. We also describe the Frobenius
automorphism for the non-equivariant limit of this equations which correspond
to the generalized Bessel equations.

### 7.2.

As before, let $T=(\mathbb{C}^{\times})^{n}$ be the torus acting on
$\mathbb{C}^{n}$ by scaling the coordinate lines with characters
$a_{1},\dots,a_{n}$. We consider the induced action on
$X=\mathbb{P}^{n-1}=\mathbb{P}(\mathbb{C}^{n})$. The equivariant cohomology
ring has the form

(7.1) $\displaystyle
H^{\bullet}_{T}(X)=\mathbb{Z}[x,a_{1},\dots,a_{n}]/(x-a_{1})\dots(x-a_{n}).$

Taking the formal limit $\hbar\to\infty$ of (6.2) we obtain the gamma class
for $X=\mathbb{P}^{n-1}$:

$\Gamma(x,\boldsymbol{a},z)=\dfrac{z^{x}}{\prod\limits_{i=1}^{n}\Gamma(x+1-a_{i})}\in\hat{H}^{\bullet}_{T}(X).$

The normalized equivariant cohomological vertex function (the $J$-function) of
$\mathbb{P}^{n-1}$ equals:

$\widetilde{V}(x,\boldsymbol{a},z)=\sum\limits_{d=0}^{\infty}\,\Gamma(x+d,\boldsymbol{a},z)\,\in\,\hat{H}^{\bullet}_{T}(X)[[z]].$

More explicitly

$\widetilde{V}(x,\boldsymbol{a},z)=\dfrac{z^{x}}{\prod\limits_{i=1}^{n}\Gamma(x+1-a_{i})}\,\sum\limits_{d=0}^{\infty}\,\dfrac{z^{d}}{(x-a_{1}+1)_{d}\dots(x-a_{n}+1)_{d}}.$

This power series satisfies the differential equation

$P(\boldsymbol{a},z)\widetilde{V}(x,\boldsymbol{a},z)=(x-a_{1})\dots(x-a_{n})\Gamma(x,\boldsymbol{a},z),$

where $P(\boldsymbol{a},z)$ is the differential operator

(7.2) $\displaystyle P(\boldsymbol{a},z)=z-(D-a_{1})\cdots(D-a_{n}),\ \ \
D=z\dfrac{d}{dz}$

and therefore in the ring (7.1) we have

$P(\boldsymbol{a},z)\widetilde{V}(x,\boldsymbol{a},z)=0.$

Formal limit $h\to\infty$ of Theorem 6.1 then gives:

######  Theorem 7.1.

For any $\boldsymbol{a}=(a_{1},\dots,a_{n})\in\mathbb{Z}^{n}_{p}$ there exists
unique Frobenius intertwiner from the differential equation (7.2) with
parameters $(a_{1},\dots,a_{n},z^{p})$ to the differential equation (7.2) with
parameters $(pa_{1},\dots,pa_{n},z)$. The constant term of the intertwiner is
represented by the operator acting in the completion (7.1) as an operator of
multiplication by the cohomology class

(7.3) $\displaystyle\mathsf{U}(0)=\prod\limits_{j=1}^{n}{\Gamma_{p}}(px-
pa_{j}).$

We note that the arguments of the gamma functions in the product (7.3) are the
$p$ multiples of the Chern roots of the tangent bundle $TX$. Thus, the
universal formula (6.7) holds, in agreement with the Remark of the previous
Section.

### 7.3.

At $\boldsymbol{a}=(0,\dots,0)$ the equivariant cohomology of
$X=\mathbb{P}^{n-1}$ specializes at the usual (non-equivariant) cohomology:

(7.4) $\displaystyle H^{\bullet}(X)=\mathbb{Z}[x]/(x)^{n}.$

The gamma class and the vertex function have the form

$\Gamma(x,z)=\dfrac{z^{x}}{\Gamma(x+1)^{n}},$

and

(7.5)
$\displaystyle\widetilde{V}(x,z)=\sum\limits_{d=0}^{\infty}\,\Gamma(x+d,z)=\dfrac{z^{x}}{\Gamma(x+1)^{n}}\,\sum\limits_{d=0}^{\infty}\,\dfrac{z^{d}}{(x+1)^{n}_{d}},$

respectively. The vertex function solves

$P(z)\widetilde{V}(x,z)=x^{n}\,\Gamma(x,z)=0$

where

(7.6) $\displaystyle P(z)=z-D^{n}$

The differential equation defined by (7.6) is the generalized Bessel equation.
For $n=2$ this is the classical Bessel equation. The components of
$\widetilde{V}(x,z)$ in any basis of cohomology (7.4) provide a basis of
solutions for these equations. In this case, the only natural basis is given
by the powers of the hyperplane class $x$:

(7.7) $\displaystyle\textrm{Basis of}\ \
H^{\bullet}(X)=\\{1,x,x^{2},\dots,x^{n-1}\\}.$

In order to compute the corresponding components it is enough to expand (7.5)
in the Taylor expansion up to degree $x^{n}$:

$\widetilde{V}(x,z)=\widetilde{V}_{0}(z)+\widetilde{V}_{1}(z)x+\dots+\widetilde{V}_{n-1}x^{n-1}$

The functions $\widetilde{V}_{0}(z),\dots,\widetilde{V}_{n}(z)$ provide a
basis of solutions to the generalized Bessel equation. Note that (7.5) has a
multiple $z^{x}$ which has the following expansion:

$z^{x}=\sum\limits_{m=0}^{n-1}\,\dfrac{\ln(z)^{m}}{m!}x^{m}$

and thus the solutions may be written in the form

(7.13)
$\displaystyle(\widetilde{V}_{0}(z),\dots,\widetilde{V}_{n-1}(z))=({V}_{0}(z),\dots,{V}_{n-1}(z))\cdot\left(\begin{array}[]{ccccc}1&\ln(z)&\ln(z)^{2}/2&..&\ln(z)^{n-1}/(n-1)!\\\
0&1&\ln(z)&..&\ln(z)^{n-2}/(n-2)!\\\ 0&0&1&..&\dots\\\ ..&..&..&..&\dots\\\
0&0&0&..&1\end{array}\right)$

where $V_{i}(z)$ are some analytic (power series in $z$) functions. In
particular, $\tilde{V}_{0}(z)$ does not contain logarithmic terms. For $n=2$,
$\tilde{V}_{0}(z)$ is the power series representing the classical Bessel
function of the first kind.

### 7.4.

When all $a_{i}=0$, the Theorem 7.3 specializes to:

######  Theorem 7.2.

The generalized Bessel equation admits Frobenius intertwiner $z\to z^{p}$ with
the constant term acting in $H^{\bullet}(X)$ as the operator of multiplication
by the cohomology class

(7.14) $\displaystyle\mathsf{U}(0)={\Gamma_{p}}(px)^{n}$

By construction, the intertwiner of Theorem 7.3 maps the solutions
$\tilde{V}(x,z^{p})$ to the solutions $\tilde{V}(px,z)$. Let
$p^{\mathrm{deg}}$ be the operator of the cohomological grading, acting on the
cohomology by $p^{\mathrm{deg}}(x^{k})=p^{k}x^{k}$. The above theorem says
that

${\Gamma_{p}}(x)^{n}\cdot p^{\mathrm{deg}}$

is the zero term of the intertwiner between $\tilde{V}(x,z^{p})$ and
$\tilde{V}(x,z)$ (note that as in (7.13), the operators act by multiplication
from the right in our conventions). In other words this gives an automorphism
of the corresponding differential equation. We thus conclude with

######  Theorem 7.3.

The generalized Bessel equation admits unique Frobenius automorphism acting by
$z\to z^{p}$. The constant term of the automorphsim acts in cohomology as the
operator

(7.15) $\displaystyle\mathsf{U}(0)={\Gamma_{p}}(x)^{n}\cdot p^{\mathrm{deg}}.$

Let us compute the matrix (7.15) explicitly in the basis (7.7). If
$\Psi(x,z)=\Psi_{0}(z)+\Psi_{1}(z)x+\dots+\Psi_{n-1}x^{n-1}$, then the
convention is that the matrices of the operators act on the row vector
$(\Psi_{0}(z),\Psi_{1}(z),\dots,\Psi_{n-1})$ by multiplication from the right.
For instance, multiplication by $x$ acts by
$x\Psi(x,z)=\Psi_{0}(z)x+\Psi_{1}(z)x^{2}+\dots+\Psi_{n-2}x^{n-1}$, and thus
the corresponding matrix is

$x=\left(\begin{array}[]{cccccc}0&1&0&0&\dots&0\\\ 0&0&1&0&\dots&0\\\
0&0&0&1&\dots&0\\\ ..&..&..&..&\dots&1\\\ 0&0&0&0&\dots&0\\\
\end{array}\right).$

Let us recall that the logarithm of the Morita gamma function has the
following Taylor expansion, see for instance Sections 58 and 61 in [Sh]:

$\log\Gamma_{p}(x)=\Gamma_{p}^{\prime}(0)x-\sum\limits_{m\geqslant
2}\,\dfrac{\zeta_{p}(m)}{m}\,x^{m}$

where $\zeta_{p}(m)$ denote the values of $p$-adic zeta function. It is known
that $\zeta_{p}(m)=0$ for even $m$, so only the odd powers of $x$ appear in
the expansion. Thus, in our conventions we obtain that the operator of
multiplication by $\log\Gamma_{p}(x)^{n}$ is given by the $n\times n$ matrix

$\log\Gamma_{p}(x)^{n}=\left(\begin{array}[]{cccccc}0&\nu_{1}&0&\nu_{3}&\dots&\dots\\\
0&0&\nu_{1}&0&\nu_{3}&\dots\\\ 0&0&0&\nu_{1}&\dots&\dots\\\
..&..&..&..&\dots&0\\\ ..&..&..&..&\dots&\nu_{1}\\\ 0&0&0&0&\dots&0\\\
\end{array}\right),$

where we denoted $\nu_{1}=n\Gamma_{p}^{\prime}(0)$ and
$\nu_{m}=-n\,\zeta_{p}(m)/m$ for $m>2$. We also obviously have

$p^{\mathrm{deg}}=\left(\begin{array}[]{cccccc}1&0&0&0&\dots&0\\\
0&p&0&0&\dots&0\\\ 0&0&p^{2}&0&\dots&0\\\ ..&..&..&..&\dots&0\\\
..&..&..&..&p^{n-2}&0\\\ 0&0&0&0&\dots&p^{n-1}\\\ \end{array}\right).$

Finally, we obtain that the constant term of the automorphism has the
following matrix:

$\mathsf{U}(0)={\Gamma_{p}}(x)^{n}\cdot
p^{\mathrm{deg}}=\exp\left(\begin{array}[]{cccccc}0&\nu_{1}&0&\nu_{3}&\dots&\dots\\\
0&0&\nu_{1}&0&\nu_{3}&\dots\\\ 0&0&0&\nu_{1}&\dots&\dots\\\
..&..&..&..&\dots&0\\\ ..&..&..&..&\dots&\nu_{1}\\\ 0&0&0&0&\dots&0\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccccc}1&0&0&0&\dots&0\\\
0&p&0&0&\dots&0\\\ 0&0&p^{2}&0&\dots&0\\\ ..&..&..&..&\dots&0\\\
..&..&..&..&p^{n-2}&0\\\ 0&0&0&0&\dots&p^{n-1}\\\ \end{array}\right)$

and the exponent of a nilpotent operator is, of course, elementary to compute.

######  Example.

For the case of the classical Bessel equation, corresponding to $n=2$, this
matrix has the form

(7.18)
$\displaystyle\mathsf{U}(0)=\left(\begin{array}[]{ll}1,&p\,\Gamma^{\prime}_{p}(0)\\\
0,&p\end{array}\right).$

The existence of the Frobenius automorphism for the Bessel equation was
discovered by B. Dwork [Dwo74]. In particular, in Lemma 3.2 of [Dwo74] he gave
an explicit formula for the matrix $\mathsf{U}(0)$ in terms of certain
converging $p$-adic series. One checks that Dwork’s matrix $\mathfrak{U}(0)$
coincides with (7.18) after transposition. In particular, the series for the
coefficients $\mathfrak{U}_{2,1}(0)=\gamma$ converges to
$p\Gamma^{\prime}_{p}(0)$.

The action of the Frobenius automorphism for the generalized Bessel equations
($n>2$) was further studied by S. Sperber in [Sp80]. The results of this
Section are in full agreement with these calculations too.

##  8\. Frobenius intertwiners at $p^{s}$ roots of unity

### 8.1.

Let $\mathsf{U}(z)$ be the Frobenius intertwiner for the $q$-difference
equation as in Theorem 3.2. As we expain in Section 3.2, the defining property
of the Frobenius intertwiner $\mathsf{U}(z)$ discussed in the previous
Sections is the condition

$\mathsf{U}(z)\pmod{p^{s}}={\mathrm{rational\ function\ of}}\ \ z$

In fact, a stronger property holds

$\mathsf{U}(z)\pmod{[p^{s}]_{q}}={\mathrm{rational\ function\ of}}\ \ z,$

where

$[p^{s}]_{q}=\dfrac{1-q^{p^{s}}}{1-q}=\varphi_{p}(q)\varphi_{p}(q^{p})\dots\varphi_{p}(q^{p^{s-1}})$

and $\varphi_{p}(q)=1+q+\dots+q^{p-1}$ is the $p$ \- th cyclotomic polynomial.
This result follows from:

######  Theorem 8.1.

For $s>0$ let $\zeta\in\mathbb{K}$ denotes a primitive $p^{s}$ \- th roots of
unity, then

$\left.\mathsf{U}(z)\right|_{q=\zeta}$

is well defined and is a rational function of $z$.

###### Proof.

At $q=\zeta$ we have the following (obvious) identity

$(z,q)_{p^{s}}=(z^{p},q^{p})_{p^{s-1}}=(1-z^{p^{s}})$

Arguing as in the proof of Lemma 4.1 we would find that at $q=\zeta$:

$\dfrac{[q^{i},q]_{p^{s}}}{[1,q]_{p^{s}}}=\left\\{\begin{array}[]{ll}\dfrac{[q^{i},q^{p}]_{p^{s-1}}}{[1,q^{p}]_{p^{s-1}}}=1,&\mathrm{if}\
\ p\mid i,\\\ &\\\ 0,&\mathrm{if}\ \ p\nmid i.\end{array}\right.$

We then use this result as the analog of Lemma 4.1 to prove that the
intertwiner (4.2) is rational, following the same steps as in the proof of
Theorem 3.2. ∎

######  Remark.

For the case $n=1$ this result was recently obtained by D. Vargas-Montoya in
[VM22]. In the terminology of [VM22] Theorem 8.1 can be formulated as
existence of the strong Frobenius structure for the $q$-hypergeometric
equation (1.2).

##  9\. Concluding remarks and future directions

### 9.1.

It is known that the Nakajima quiver varieties have very rich enumerative
geometry [Oko17]. Partition functions of quasimaps to the Nakajima quiver
varieties with various boundary conditions satisfy remarkable systems of
$q$-difference equations, known as the quantum difference equations [OS22]. In
the present paper we discuss the existence of the Frobenuis structure for the
quantum difference equations associated with simplest Nakajima variety
$X=T^{*}\mathbb{P}^{n-1}$. All essential ingredients in our construction,
e.g., the existence of the rational shift operators in the equivariant
parameter $\hbar$, exist for general Nakajima varieties too. For this reason,
the $p$ \- adic quantum difference equations of the Nakajima varieties are
equipped with the Frobenius structure. We plan to return to the case of
general $X$ in a separate note. In the cohomological limit $q\to 1$ the
quantum difference equations degenerate to the quantum differential equations
for the Nakajima varieties [MO19], therefore the Frobenius structures exists
for such differential equations also.

### 9.2.

The quantum difference/differential equations of the Nakajima varieties are
connected deeply with representation theory of quantum groups. It is an
interesting problem to find a representation theoretic description of the
Frobenius intertwiners for these equations. In particular, the quantum
difference equations for $q$ specialised at the roots of unity are controlled
by the Bethe ansatz, see [AO17, PSZ, KPSZ]. It seems possible to describe the
Frobenius intertwiners for these values of $q$ as transition matrices between
various Bethe eigenvectors. We plan to investigate this question further in
[KS].

### 9.3.

In the framework of enumerative geometry, one can define the Frobenius
intertwiner as the partition function counting quasimaps from $\mathbb{P}^{1}$
to a Nakajima variety $X$ with two different relative conditions at
$0\in\mathbb{P}^{1}$ and $\infty\in\mathbb{P}^{1}$. The corresponding moduli
space of quasimaps is represented by Fig.1, we refer to [Oko17] and also [PSZ]
where the pictorial notations are explained.

Figure 1. Frobenius intertwiner as a partition function of relative quasimaps

At $0\in\mathbb{P}^{1}$ we have the usual relative condition on quasimaps as
defined in [Oko17]. We recall that the relative boundary condition at
$0\in\mathbb{P}^{1}$ means that $\mathbb{P}^{1}$ is allowed to have a chain of
rational “bubbles” with simple nodal intersections. This chain of bubbles is
represented by blue color in Fig. 1.

At $\infty\in\mathbb{P}^{1}$ we also impose a relative condition but require,
in addition, the relative quasimaps to be $\mathbb{Z}/p\mathbb{Z}$-invariant.
More precisely, recall that the moduli space of relative quasimaps is defined
as a stack quotient of quasimaps from a chain of $\mathbb{P}^{1}$’s by the
automorphism group $(\mathbb{C}^{\times})^{l}$ where $l$ is the length of the
chain, see Section 6.2 in [Oko17]. The moduli space of relative quasimaps at
$\infty\in\mathbb{P}^{1}$ is constructed as follows: first we consider the
space of quasimaps from a chain of rational curves of length $l$ which are
invariant with respect to $\mathbb{Z}/p\mathbb{Z}$, which acts in each
“bubble” (the component of the chain) via multiplication by $p$ \- th roots of
unity. Then we define the $\mathbb{Z}/p\mathbb{Z}$-invariant quasimaps as the
stack quotient of this moduli space by $(\mathbb{C}^{\times})^{l}$.

It is known, Theorem 8.1.16 in [Oko17], that the partition function of
relative quasimaps from the chain of bubbles at $0\in\mathbb{P}^{1}$ is a
fundamental solution matrix $\Psi({\boldsymbol{u}},\hbar,q,z)$ of the quantum
difference equation associated with $X$, normalized so that

$\Psi({\boldsymbol{u}},\hbar,q,z)=1+O(z),$

in terminology of [Oko17] $\Psi({\boldsymbol{u}},\hbar,q,z)$ is also referred
to as the capping operator.

Similarly, the partition function of $\mathbb{Z}/p\mathbb{Z}$ \- invariant
quasimaps at $\infty\in\mathbb{P}^{1}$ can be shown to be equal to
$\Psi({\boldsymbol{u}},\hbar,q^{p},z^{p})^{-1}$. Finally, the contribution of
degree $0$ \- quasimaps which correspond to the “constant” quasimaps from the
parameterized component $\mathbb{P}^{1}$ can be computed by
$\mathbb{C}^{\times}_{q}$ \- equivariant localization similarly to how it was
done for the shift operators in Section 8.2 of [Oko17], in particular see
Lemma 8.2.12. The resulting contribution has the form

$\mathsf{U}(0)=\Gamma_{p,q}\Big{(}(q-\hbar)T^{1/2}X\Big{)},$

where $T^{1/2}X$ denotes the corresponding choice of the polarization bundle
of $X$. For instance, in the case $X=T^{*}\mathbb{P}^{1}$ this formula
specializes to (3.5). Combining all this together we obtain that the
corresponding partition function is

$\mathsf{U}(z)=\Psi({\boldsymbol{u}},\hbar,q,z)\mathsf{U}(0)\Psi({\boldsymbol{u}},\hbar,q^{p},z^{p})^{-1}$

which is exactly the Frobenius intertwiner.

We note that the idea of $\mathbb{Z}/p\mathbb{Z}$ \- invariant quasimap moduli
is similar to the moduli space recently used in [HL24] to define the quantum
Steenrod operators in quantum cohomology.

### 9.4.

Frobenius intertwiners link the theory of $p$ \- adic differential equations
with the theory of exponential sums in number theory [K90]. Namely,
interesting sums arise as the traces

(9.1) $\displaystyle\textrm{Tr}(\mathsf{U}(T_{z}))$

where $T_{z}\in\mathbb{Q}_{p}$ denotes the Teichmuller representative of an
element $z\in\mathbb{F}_{p}$. These sums may be considered as the arithmetic
counterpart of the integral representation for solutions of the corresponding
differential equation. See for instance [Dwo74], where the Kloosterman sum is
realized as the trace of Frobenius for the Bessel differential equation.

These type of sums for the $q$ \- case seem never considered. One interesting
new feature of the $q$ \- difference equations for Nakajima varieties is the
$3D$-mirror symmetry, which is the invariance of the equations with respect to
transformation $z\longleftrightarrow{\boldsymbol{u}}$, where $z$ denote the
Kähler parameters (parameters counting the degrees of the quasimaps) and
${\boldsymbol{u}}$ denote the equivariant parameters. It is interesting to
investigate what part of this symmetry survives in the arithmetic context of
the exponential sums (9.1).

## Acknowledgements

I thank K. Kedlaya, J. Hee Lee, A. Oblomkov, D. Vargas-Montoya, M. Vlasenko
and especially A. Okounkov for very inspiring discussions. I thank A.
Varchenko for introducing me to the subject. This note is the extended version
of the talks [Sm23] and [Sm24], I thank the organizers for these
opportunities. This work is supported by NSF grants DMS - 2054527 and
DMS-2401380.

## References

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# PROBING STATISTICAL REPRESENTATIONS FOR END-TO-END ASR

###### Abstract

End-to-End automatic speech recognition (ASR) models aim to learn a
generalised speech representation to perform recognition. In this domain there
is little research to analyse internal representation dependencies and their
relationship to modelling approaches. This paper investigates cross-domain
language model dependencies within transformer architectures using SVCCA and
uses these insights to exploit modelling approaches. It was found that
specific neural representations within the transformer layers exhibit
correlated behaviour which impacts recognition performance.

Altogether, this work provides analysis of the modelling approaches affecting
contextual dependencies and ASR performance, and can be used to create or
adapt better performing End-to-End ASR models and also for downstream tasks.

Index Terms—  speech recognition, end-to-end, cross domain, transformer,
analysis, language modelling

##  1 Introduction

The typical approach to develop a framework for ASR has been to use deep
neural networks to recognise and align acoustic features to graphemes or
phonemes; replacing the requirement for distinctly-optimised modules, such as
acoustic, pronunciation or language models (LMs). Using End-to-End modelling
approaches reduces the need for expert domain knowledge as it aims to jointly
optimise the training regime while adapting to diverse speech environments.
These factors have led to End-to-End ASR models becoming a popular choice for
on-device deployment.

Current research in End-to-End ASR modelling is dominated by three approaches:
recurrent-transducers [1], Connectionist Temporal Classification models [2] or
attention-based encoder-decoder architectures [3, 4]. End-to-End ASR
frameworks are typically dependant on the amount of data resources and are
commonly fine-tuned to the corpora in order to improve the recognition
performance, which has directed techniques to improve the representation
capacity of the modelling approaches [5, 6]. The ability to use larger amounts
of training data significantly improves the recognition performance of End-to-
End models [7]. However, previous work [8] suggested that the relationship
between memorisation and generalisation within these networks remains elusive,
being referred to as a “black-box". It can be hypothesised that richer neural
representations are not analogous to increasing neural depth or model size
[9]. By learning representations of speech that are robust, the general
recognition performance of the model should improve proportionately without
the requirement for increasing model size or training data.

As the development of End-to-End ASR models continues, there is also a
corresponding demand to be able to interpret the internal latent
representations and the explainability of modelling approaches, such as the
potential relationship between training data and the integration of LMs.
Despite numerous variations of modelling approaches, there has also been
little exploration of the internal model representations and their
relationship to model recognition performance across different domains. Layer-
wise analysis of models has been used to interpret modelling approaches and
relationships between representations in multiple domains [10, 11, 12]. SVCCA
analysis techniques have been used to highlight neural representations with
respect to their ability to generalise, by observing the relationship between
the correlation coefficients of neural layers during training [13].

As transformer modelling approaches achieve state-of-the-art results for End-
to-End ASR, this work aims to understand how representations in transformer
models adapt to out-of-domain LMs by analysing how the representations evolve
across layers. The relationship between ASR performance and the neural
representations is explored. Using unmatched sub-word LMs, it is possible to
observe the dependencies of the representations within model layers and
observe the impact of the variations. Section 5 shows that observing the
representation dependencies is important to develop intuitive modelling
approaches and improve recognition performance.

The analysis methodology is first defined in Section 2, then the developed
framework for analysis experiments is described in Section 3.3. The
implementation of cross-domain LMs in the End-to-End ASR models is defined in
Section 3.2 and the analysis experiments are shown in Section 4. Experiments
analysing the adaptation of transformer model parameters are conducted in
Section 5 with the results discussed in further detail in Section 6. It was
found that deeper transformer model layers contain learned representation
dependencies for cross-domain LMs and recognition performance can be improved
by tuning the parameters to the hierarchical dependencies.

##  2 Correlation Analysis Methodology for End-to-End ASR Models

To map the posterior distribution of $p(Y|X)$ jointly, End-to-End ASR
modelling approaches aim to approximate function of acoustic input
$\mathcal{X}=\\{x_{1},...,x_{T}\\}$ of length $\mathcal{T}$ over output labels
$\mathcal{Y}=\\{y_{1},...,y_{N}\\}$. The dependencies of the End-to-End model
parameters, are not as clearly distinguished as traditional modular approaches
to acoustic and language modelling, as they are not separately defined. These
“black-box" models are complex and it can be difficult to understand the
relationship between the internal parameter dependencies and the resulting
performance across datasets. By using SVCCA analysis from [14], it is possible
to observe a relationship between two sets of observations by the maximum
correlations between the linear projections. This correlation analysis method
is used to measure the linear relationship between vectors representing neural
layers. For ASR models, the activation output of a layer
$z^{l}_{i}=\\{z_{i}^{l}(x_{1}),...,(z_{i}^{l}(x_{N})\\}$ are extracted for
dataset $\mathcal{X}$ and neuron $i$ in layer $l$. To observe the similarity
between or within neural layers and to compare the relationship of the
internal representation dependencies upon different variables, the use of
different LMs and model parameters are varied and the similarities between
layer representations are compared to derive insights regarding their
relationship.

To measure correlation for $\mathcal{N}$ datapoints, pairs of vectors are
sampled from views of layers $l_{1}$ and $l_{2}$, which are projected, where
$l_{1}=\\{z_{1}^{l_{1}},...,z_{N_{1}}^{l_{1}}\\}$ and
$l_{2}=\\{z_{2}^{l_{2}},...,z_{N_{2}}^{l_{2}}\\}$. The projected views of
$l_{1}$ and $l_{2}$ are pruned by the application of singular vector
decomposition (SVD) to retain 99% of the representative dimensions and to
reduce the impact of potential noise. The application of SVD forms subspaces
$l_{1}^{\prime}\subset l_{1}$ and $l_{2}^{\prime}\subset l_{2}$ and then CCA
can then be applied to find vectors $v$ and $s$ that maximise correlation
$\rho$ between the projections of $l_{1}^{\prime}$ and $l_{2}^{\prime}$:

$\rho=\frac{\langle
v^{T}l_{1}^{\prime},s^{T}l_{2}^{\prime}\rangle}{||v^{T}l_{1}^{\prime}||\;||s^{T}l_{2}^{\prime}||}$
(1)

where $v,\;s$ are transforms that aim to maximise the correlation of the
vectors. $\rho$ increases where neural representations have encoded more
similar information. Further details regarding SVCCA analysis can be found in
[14].

##  3 Experimental Setup

###  3.1 Transformers for ASR

Transformers, initially published in [3], are a widely chosen encoder-decoder
architecture for speech recognition frameworks due their ability to
parallelise the training regime. This enables use of larger amounts of
training data. The encoder structure is comprised of stacked self-attention
and point-wise, fully connected layers in blocks, with each block managed by a
multi-head self-attention layer and feed forward layer. The attention module
is a major component of the architecture as it enables pairwise position
measurement for windows of the input speech. After the attention vector has
passed through the feed forward layer, the output is passed onto another self-
attention layer within the decoder to compute the embedding vectors. A final
attention mechanism attends over the embedding vectors, in order to compute
the relationship between the input and output sequences, before being passed
to the final linear unit. A softmax function is then calculated over the
target output.

A CNN front-end is incorporated with the transformer layers for feature
extraction. The final convolutional layer was then projected to 12 stacked
transformer encoder blocks with embedding dimensions of $512\times 2048$ and 6
decoder layers with positional embeddings.

###  3.2 Language Modelling in End-to-End ASR

The integration of LMs within End-to-End architectures is relatively common-
place as they can be used to supervise the training optimisation and also for
decoding to aid recognition performance [15, 16]. However, it is unclear how
the internal dependencies of the End-to-End models handle latent LM
representations and whether there are similar learned representation spaces
that are robust across different domains. By training models with cross-domain
LMs, it is possible to observe these dependencies by comparing models with
SVCCA analysis.

For the analysis experiments, a sub-word LM is integrated by shallow-fusion
decoding [17] and label smoothing [18] techniques. The sub-word LM is a 3
layer LSTM model. Shallow-fusion decoding computes the weighted sum of a pair
of posterior distributions over sub-words; using one from the ASR model and
one from the sub-word LM. The sub-word LM is an LSTM-based LM trained with
restricted computational complexity, by only keeping the most frequent sub-
words and splitting the rest into characters, to enable conversion with low
information loss. Label smoothing computes the cross entropy loss during the
model’s training regime with a weighted mix of distributions from a unigram LM
and one-hot targets from the dataset.

###  3.3 Correlation Analysis Framework

The framework developed in [12] was utilised to investigate the relationships
between internal dependencies and save the models during training. For all the
experiments, the transformer models and LMs were trained using the ESPRESSO
framework [16]. The analysis was conducted for all models by extracting the
activation outputs of each neural layer of the encoder for each training
epoch. Each model was saved throughout all epochs and then a controlled input
of 100 frames of unseen speech data was fed through the layers, whilst
simultaneously extracting the activation outputs for each layer.
80-dimensional log Mel acoustic features with additional pitch features were
extracted, from 25ms windows with a stride of 10ms, for all models.

####  3.3.1 Data

For the experiments, three common US-English datasets from differing domains
for ASR were chosen: Switchboard [19], Librispeech [20] and WSJ [21]. The
Switchboard dataset contains conversational telephone speech, Librispeech is a
compilation of read audiobooks, and WSJ contains read news. The test sets for
the Switchboard dataset, referred to as Swbd and Callhome, are derived from
the LDC2002S09 set and contain 20 unreleased telephone conversations from
Switchboard and 20 telephone unscripted conversations from Callhome. To ensure
the transformer models were converged by training on comparative data to the
LM, the model trained with Switchboard used up-sampled data (to 16kHz). The
LMs trained with Librispeech were trained using the full 960 hour training set
and the ASR models were tested on the test-clean and test-other sets. The
training set for the WSJ LM was the si284 set, with the Dev93 set for
validation and Eval92 for testing the ASR model performance.

##  4 Cross-domain Language model Experimentation

Correlation analysis of the neural representations across the transformer
model layers is used to measure and analyse the changes in correlation when
cross-domain LMs are integrated. Figure 1 shows the SVCCA coefficients, as
training converges, between the encoder layers of two transformer models. The
models were trained with Switchboard data but one model uses an in-domain
Fisher sub-word LM, and the other model uses an out-of-domain WSJ sub-word LM.
These models are both trained with sub-word units using SentencePiece [22] and
integrated during the training process using the scheduled sampling method and
decoded with shallow-fusion, as described in Section 3.2. The correlation
analysis shows very little difference in coefficiency between layers 1 to 6
(top graph of Figure 1), aside from in the initial epochs which could be
attributed to the random initialisation of parameters. This suggests that the
neural layers of both of these models are converging to similar representation
spaces. However, between layers 7 to 12 (bottom graph of Figure 1, the
differences in coefficiency are much larger throughout training. This suggests
that the representations learned in these deeper layers are more dependent
upon the LM domain.

The bottom graph in Figure 2 displays the standard deviations of the
coefficiency between the models trained with cross-domain LMs. This aims to
show the variation in coefficiency by layer more clearly, where the standard
deviations in layers 10, 11 and 12 are highest. The top graph of Figure 2
shows the variance in coefficiency within the neural layers of a model trained
without scheduled sampling or shallow-fusion decoding compared to the model
trained with the Fisher sub-word LM. This suggests that a similar observation
can be made for LM specific representations, whereby the variance is higher
overall and the coefficiency of layers 8 to 12 deviates the most. The results
in Figure 2 also imply that layers 1 to 4 have very little dependency on LM
representations. These insights suggest that encoder layers 1 to 4 of the
transformer model can be frozen when fine-tuning with LMs and the optimisation
regime of End-to-End ASR models can be adapted to improve downstream tasks.

Fig. 1: SVCCA correlation coefficients as performance converges within
transformer layers 1 to 6 (top) and layers 6 to 12 (bottom), between a model
trained with a Fisher-based LM and a model trained with a WSJ-based LM

Fig. 2: Standard deviation of correlations across transformer model layers
with and without a LM (top) and with cross-domain LMs (bottom)

Regarding performance, the model that was trained with the Fisher LM reached
9.5% word error rate (WER) on the Switchboard test set and 19.1% on the
Callhome test set, while the model that was trained with the WSJ LM was 10.7%
and 21.1% respectively. The differences in recognition performance are
attributed to the domains of the LMs and the test sets used for evaluation.

##  5 Modelling Structure Analysis

To optimise the parameters for state-of-the-art End-to-End ASR models, many
iterations are trained with slight parameter modifications. Optimisation of
model parameters to specific datasets to achieve the best recognition
performance possible [16, 15] is referred to here as tuned. For example, the
dimensionality, number of layers and also the hyperparameters have been
observed to impact the recognition performance. As shown in Table 1, using a
transformer model with the same parameters and composition for several
datasets does not achieve the lowest WER across all of the datasets. These
tuned models are typically reached by extensive hyperparameter optimisation
techniques, which are computationally expensive and considerably time
consuming without providing observational evidence regarding the dependencies
of certain parameters upon the recognition performance.

Using cross-corpora correlation analysis while varying the parameters, it is
possible to interpret these dependencies in a more meaningful way and provide
some observational evidence to reduce the future need for extensive
hyperparameter optimisation when developing new models or fine-tuning trained
models. By understanding the representation dependencies, this can potentially
reduce the computational resources required to improve speech recognition
model architectures. Table 1 shows the results of 3 transformer models with
variations in model parameters that are used in state-of-the-art End-to-End
ASR frameworks. All models are the same transformer-based encoder-decoder
architecture with the following variations:

  * •

Model 1 has an embedding dimension of 512, a feed forward embedding dimension
of 2048, 4 attention heads, and an attention dropout of 0.25.

  * •

Model 2 has an embedding dimension of 256, a feed forward embedding dimension
of 1024, 4 attention heads, and attention dropout of 0.25.

  * •

Model 3 has an embedding dimension of 512, a feed forward embedding dimension
of 2048, 8 attention heads, and an attention dropout of 0.1.

To observe the relationship between the learned representations of the adapted
models and attribute these adaptations to improved recognition performance
with specific data, the model performance was assessed across all test sets,
as shown in Table 1. For the Switchboard and Callhome test sets, the
recognition performance of model 1 is the best, while model 2 reaches slightly
worse performance on the Callhome set and model 3 has the highest WER for both
test sets.

Table 1: Transformer model WER on EVAL’00, WSJ and Librispeech test sets with
tuned parameters

Model | Swbd | Chm | Eval92 | Dev93 | Test-cln | Test-oth
---|---|---|---|---|---|---
M1 | 9.5 | 19.1 | 4.59 | 7.54 | 3.5 | 8.51
M2 | 9.6 | 20 | 4.13 | 6.3 | 3.99 | 8.72
M3 | 10.4 | 21.6 | 4.52 | 7.43 | 1.9 | 3.9

Figure 3 displays the SVCCA coefficients for each model trained with the
Switchboard dataset. Model 2’s mean coefficiency, across layerwise
representations, are substantially less correlated than the other models. The
standard deviations of the correlations within these layers also vary
significantly higher than Model 1 or 3. Model 3’s mean coefficiency across
layerwise representations is fairly similar for all layers with very small
standard deviation. It is observed that correlations within the layers of
model 2 have lower coefficiency, and the recognition performance of this model
is lower for the Switchboard test sets. Also, there are little hierarchical
coefficiency patterns throughout the layers of model 3, and this model also
has a slightly worse performance, which corroborates with results from [12].
Model 1 has lower coefficiency within layers 8-12 and has the best recognition
performance.

Fig. 3: Transformer SVCCA coefficients as performance converges in model 1
(top), model 2 (middle) and model 3 (bottom) trained with Switchboard data

##  6 Discussion

The findings in Section 3.2 correlate with findings from [23] where semantic
and syntax level features of speech are predominantly dependent upon deeper
layers of transformer-based models, while acoustic and fluency features are
predominantly dependent on the shallower layers. In the case of End-to-End
transformers for ASR tasks, the LM-dependent representations are shown to be
primarily dependent within layers 7-12. The cross-domain LM-dependent
representations are observed within layers 10-12. Further experiments training
with the WSJ dataset with Fisher and WSJ sub-word LMs showed very similar
observed behaviours across layer coefficiency. These observations can be used
to check for possible biases in the modelling process that affect recognition
performance, without the need for extensive training requirements, to improve
joint optimisation. The analysis also aids in the interpretability of the
impact of representation dependencies within End-to-End ASR models.

The experiments in Section 5, attempt to show these internal dependencies with
regard to the model parameters within the same modelling approaches. As shown
in Figure 3, Model 2 used shallower embedding dimensions than model 1, which
has caused the coefficiency of many of the layers to become highly
uncorrelated. Model 3 is observed to have very highly correlated layers,
however there are little distinct hierarchies in the neural representations
when the attention heads are increased to 8 and the attention dropout is
reduced. By adapting the parameters of transformer models, the layers with the
most dependency for representing domain-specific information are altered.
These changes in hierarchical representations have been observed to impact
recognition performance, and further suggests a relationship between
correlated hierarchical representations and the ability for the model to
generalise, particularly for cross-domain speech recognition. Increasing the
attention dropout is theorised to improve model robustness [6], where typical
features of conversational speech are boundary uncertainties and hesitations.
In the case of End-to-End conversational speech recognition, the results show
that using substantial attention dropout in transformer models is important to
produce correlated hierarchies in dependent layers but also utilise a model
with sufficient embedding dimensionality that the representations within
context-critical layers don’t become too uncorrelated.

##  7 Conclusion

Using SVCCA as a correlation index has highlighted several aspects of the
relationships between the neural representations, transformer-based modelling
parameters and the impact these have upon recognition performance.
Interpretative analysis is important to develop future modelling approaches
for meaningful improvement strategies. Expanding the scope of the
investigation into the attributes and potential learned features that could be
classified within the layers would provide a deeper understanding of the
properties of these dependencies and how these could be further exploited. The
insights into the dependencies of the neural layers can be used for the
development of models for few-shot learning and downstream tasks for End-to-
End ASR.

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