# Generalized $q$-difference equations for general $q$-polynomials with double
$q$-binomial coefficients

Jian Cao1, Sama Arjika2,3 and Mahouton Norbert Hounkonnou3 missing

###### Abstract.

In this paper, we use the generalized $q$-polynomials with double $q$-binomial
coefficients and homogeneous $q$-operators [J. Difference Equ. Appl. 20
(2014), 837–851.] to construct $q$-difference equations with seven variables,
which generalize recent works of Jia et al [Symmetry 2021, 13, 1222.]. In
addition, we derive Rogers formulas, extended Rogers formulas and
Srivastava–Agarwal type bilinear generating functions for generalized
$q$-polynomials, which generalize generating functions for Cigler’s
polynomials [J. Difference Equ. Appl. 24 (2018), 479–502.]. Finally, we also
derive mixed generating functions using $q$-difference equations.

###### Key words and phrases:

Basic (or $q$-) hypergeometric series; Homogeneous $q$-difference operator;
Double $q$-binomial coefficients; Cigler polynomials; Generating functions;
Rogers type formulas; Srivastava-Agarwal type bilinear generating functions.

1School of Mathematics, Hangzhou Normal University, Hangzhou City, Zhejiang
Province, 311121, China. 2Department of Mathematics and Informatics,
University of Agadez, Niger. 3International Chair in Mathematical Physics and
Applications (ICMPA-UNESCO Chair), University of Abomey-Calavi, 072 BP 50, ,
Cotonou, Benin Republic.

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
2010 Mathematics Subject Classification.Primary: 05A30, 33D15, 33D45;
Secondary: 05A40, 11B65.

##  1\. Introduction

In this paper, we adopt the common notation and terminology for basic
hypergeometric series as in Refs. [10, 15]. Throughout this paper, we assume
that $q$ is a fixed nonzero real or complex number and $|q|<1$. The
$q$-shifted factorial and its compact factorial are defined [10, 15],
respectively by:

$(a;q)_{0}:=1,\quad(a;q)_{n}{\color[rgb]{1,0,0}:}=\prod_{k=0}^{n-1}(1-aq^{k}),\;(a;q)_{\infty}:=\prod_{k=0}^{\infty}(1-aq^{k})$
(1.1)

and
$(a_{1},a_{2},\ldots,a_{r};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\cdots(a_{r};q)_{n},\;n\in\\{0,1,2\cdots\\}$.
We will use frequently the following relation

$(aq^{-n};q)_{n}=(q/a;q)_{n}(-a)^{n}q^{-n-({}^{n}_{2})}.$ (1.2)

The generalized $q$-binomial coefficient is defined as [10]

${\,\alpha\,\atopwithdelims[
]\,k\,}_{q}=\frac{(q^{-\alpha};q)_{k}}{(q;q)_{k}}(-1)^{k}q^{\alpha
k-({}^{k}_{2})},\quad\,\alpha\in\mathbb{C}.$ (1.3)

Similarly, by replacing $q$ by $-q$ in the above relation, we get the
following:

${\,\alpha\,\atopwithdelims[
]\,k\,}_{-q}=\frac{(-q^{-\alpha};q)_{k}}{(-q;q)_{k}}q^{\alpha
k-({}^{k}_{2})},\quad\,\alpha\in\mathbb{C}.$ (1.4)

The basic or $q$-hypergeometric function in the variable $z$ (see Slater [21,
Chap. 3], Srivastava and Karlsson [24, p. 347, Eq. (272)] for details) is
defined as:

${}_{r}\Phi_{s}\left[\begin{array}[]{r}a_{1},a_{2},\ldots,a_{r};\\\ \\\
b_{1},b_{2},\ldots,b_{s};\end{array}q;z\right]=\sum_{n=0}^{\infty}\Big{[}(-1)^{n}q^{({}^{n}_{2})}\Big{]}^{1+s-r}\,\frac{(a_{1},a_{2},\ldots,a_{r};q)_{n}}{(b_{1},b_{2},\ldots,b_{s};q)_{n}}\frac{z^{n}}{(q;q)_{n}},$

when $r>s+1$. Note that, for $r=s+1$, we have:

${}_{r+1}\Phi_{r}\left[\begin{array}[]{r}a_{1},a_{2},\ldots,a_{r+1};\\\ \\\
b_{1},b_{2},\ldots,b_{r};\end{array}q;z\right]=\sum_{n=0}^{\infty}\frac{(a_{1},a_{2},\ldots,a_{r+1};q)_{n}}{(b_{1},b_{2},\ldots,b_{r};q)_{n}}\frac{z^{n}}{(q;q)_{n}}.$

We remark in passing that, in a recently-published survey-cum-expository
review article, the so-called $(p,q)$-calculus was exposed to be a rather
trivial and inconsequential variation of the classical $q$-calculus, the
additional parameter $p$ being redundant or superfluous (see, for details,
[22, p. 340]).

Chen et al. [8] introduced homogeneous $q$-difference operator $D_{xy}$ as

$D_{xy}\big{\\{}f(x,y)\\}:=\frac{f(x,q^{-1}y)-f(qx,y)}{x-q^{-1}y},$ (1.5)

which turn out to be suitable for dealing with the Cauchy polynomials.

Wang and Cao [26] presented two extensions of Cigler’s (see [3]) polynomials

$\mathcal{C}_{n}^{(\alpha-n)}(x,y,b)=\sum_{k=0}^{n}(-1)^{k}q^{({}^{k}_{2})}{\alpha\,\atopwithdelims[
]\,k\,}_{q}b^{k}\frac{(q;q)_{n}}{(q;q)_{n-k}}p_{n-k}(x,y)$ (1.6)

and

$\mathcal{D}_{n}^{(\alpha-n)}(x,y,b)=\sum_{k=0}^{n}q^{({}^{k}_{2})}{\alpha\,\atopwithdelims[
]\,k\,}_{q}b^{k}\frac{(q;q)_{n}}{(q;q)_{n-k}}\left[(-1)^{n+k}q^{-({}^{n}_{2})}p_{n-k}(y,x)\right],$
(1.7)

where $\displaystyle
p_{n}(x,y):=(x-y)(x-qy)\cdots(x-q^{n-1}y)=(y/x;q)_{n}\,x^{n}$ are the Cauchy
polynomials.

Recently, Jia et al [14] introduced the following

$L_{\tilde{m},\tilde{n}}(\alpha,x,z,a)=\sum_{k=0}^{n}{n\,\atopwithdelims[
]\,k\,}_{q}{\alpha\,\atopwithdelims[
]\,k\,}_{-q}q^{\tau(\tilde{m},\tilde{n})+({}^{k}_{2})}(a;q)_{k}z^{k}x^{n-k},$
(1.8)

with

$\tau(\tilde{m},\tilde{n})=\tilde{m}({}^{k}_{2})-\tilde{n}({}^{k+1}_{\;\,\;2}),$
(1.9)

where $\tilde{m}$ and $\tilde{n}$ are real numbers.

Our present investigation is essentially motivated by the earlier works by Jia
et al [14]. Our aim here is to introduce and study some further extensions of
the above-mentioned $q$-polynomials

$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)=\sum_{k=0}^{n}{n\,\atopwithdelims[
]\,k\,}_{q}{\alpha\,\atopwithdelims[
]\,k\,}_{-q}q^{\tau(\tilde{r},\tilde{s})+({}^{k}_{2})}(a;q)_{k}p_{n-k}(x,y)z^{k},$
(1.10)

where $\tau(\tilde{r},\tilde{s})$ is defined as in (1.9).

The main task in this paper is to show how the $q$-polynomials
$(\ref{dedref03})$ are related to other known polynomials. For example, the
generalized $q$-polynomials
$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ defined in
$(\ref{dedref03})$ are generalized and unified forms of Hahn polynomials and
Al-Salam-Carlitz polynomials. Particular cases of known results in the
literature are given in Remark 1.

######  Remark 1.

  1. (1)

Upon setting $y=0$, the generalized $q$-polynomials
$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ defined in
$(\ref{dedref03})$ reduce to (1.8) (see [14]):

$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,0,z,a)=L_{\tilde{r},\tilde{s}}(\alpha,x,z,a).$
(1.11)

  2. (2)

For $(\alpha,\tilde{r},\tilde{s},x,y,z,a)=(\infty,0,0,y,x,-z,-q)$, the
generalized $q$-polynomials
$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ reduce to trivariate
$q$-polynomials $F_{n}(x,y,z;q)$ [1]

$\tilde{L}_{n}^{(0,0)}(\infty,y,x,-z,-q)=(-1)^{n}q^{({}^{n}_{2})}F_{n}(x,y,z;q).$
(1.12)

  3. (3)

Upon setting $\alpha=n\in\mathbb{Z}$ and
$(\tilde{r},\tilde{s},a,x,y,z)=(0,-1,-yq,1,0,x)$, the $q$-polynomials
$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ reduce to
$\rho_{e}(n,y,x,q)$ (see [14]):

$\tilde{L}_{n}^{(0,-1)}(n,1,0,x,-qy)=\rho_{e}(n,y,x,q).$ (1.13)

  4. (4)

Upon setting $(\alpha,\tilde{r},\tilde{s},y,a)=(\infty,-1,0,1,-q)$, the
generalized $q$-polynomials
$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ reduce to the
homogeneous Rogers-Szegö polynomials $h_{n}(x,y|q)$ (see [18]):

$\tilde{L}_{n}^{(-1,0)}(\infty,x,y,1,-q)=h_{n}(x,y|q).$ (1.14)

  5. (5)

By choosing $(\alpha,\tilde{r},\tilde{s},a,x,y)=(\infty,-1,0,-q,xq^{-n},0)$,
the generalized $q$-polynomials
$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ reduce to the Rogers-
Szegö polynomials $g_{n}(z,x|q)$ (see [2]):

$\tilde{L}_{n}^{(-1,0)}(\infty,xq^{-n},0,z,-q)=g_{n}(z,x|q).$ (1.15)

The rest of paper is organized as follows. In section 2, we deduce the main
results of $q$-difference equations with seven-variable for generalized
$q$-polynomials. In section 3, we obtain the generating function of
generalized $q$-polynomials by the method of $q$-difference equations. In
section 4, we derive Rogers formula for generalized $q$-polynomials by using
the $q$-difference equations. In section 5, we gain a mixed generating
function for generalized $q$-polynomials by $q$-difference equations.

##  2\. Main results

In this section, we give the following fundamental theorem.

######  Theorem 2.

Let $f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$ be a seven-variable analytic
function in a neighborhood of
$(\alpha,x,y,a,z,\tilde{r},\tilde{s})=(0,0,0,0,0,0,0)\in\mathbb{C}^{7}.$ Then,
$f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$ can be expanded in terms of
$\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ if and only if $f$
satisfies the following $q$-difference equation:

$\displaystyle(x-q^{-1}y)\Big{\\{}f(\alpha,x,y,a,z,\tilde{r},\tilde{s})-f(\alpha,x,y,a,q^{2}z,\tilde{r},\tilde{s})\Big{\\}}$
$\displaystyle\quad\qquad\qquad\qquad\qquad=q^{\alpha-\tilde{s}}z\Big{\\{}f(\alpha,qx,y,a,zq^{\tilde{r}-\tilde{s}},\tilde{r},\tilde{s})-f(\alpha,x,q^{-1}y,a,zq^{\tilde{r}-\tilde{s}},\tilde{r},\tilde{s})\Big{\\}}$
$\displaystyle\quad\qquad\qquad\qquad\qquad+q^{-\tilde{s}}(1-aq^{\alpha})\Big{\\{}f(\alpha,qx,y,a,zq^{1+\tilde{r}-\tilde{s}},\tilde{r},\tilde{s})-f(\alpha,x,yq^{-1},a,zq^{1+\tilde{r}-\tilde{s}},\tilde{r},\tilde{s})\Big{\\}}$
$\displaystyle\quad\qquad\qquad\qquad\qquad-
azq^{-\tilde{s}}\Big{\\{}f(\alpha,qx,y,a,zq^{2+\tilde{r}-\tilde{s}},\tilde{r},\tilde{s})-f(\alpha,x,yq^{-1},a,zq^{2+\tilde{r}-\tilde{s}},\tilde{r},\tilde{s})\Big{\\}}.$
(2.1)

######  Remark 3.

For $y=0$ in Theorem 2, we get the concluding remarks of Jia et al [14].

To determine whether a given function is an analytic function in several
complex variables, we often use the following Hartogs’s theorem. For more
information, please refer to [16, 25].

######  Lemma 4 ([11, Hartogs’s theorem]).

If a complex-valued function is separately holomorphic (analytic) in each
variable in an open domain $D\in\mathbb{C}^{n}$, then it is holomorphic
(analytic) in $D$.

In order to prove Theorem 2, we need the following fundamental property of
several complex variables.

######  Lemma 5 ([17, Proposition 1]).

If $f(x_{1},x_{2},...,x_{k})$ is analytic at the origin
$(0,0,...,0)\in\mathbb{C}^{k}$, then, $f$ can be expanded in an absolutely
convergent power series,

$\displaystyle
f(x_{1},x_{2},...,x_{k})=\sum_{n_{1},n_{2},...,n_{k}=0}^{\infty}\alpha_{n_{1},n_{2},...,n_{k}}x_{1}^{n_{1}}x_{2}^{n_{2}}...x_{k}^{n_{k}}.$

###### Proof of Theorem 2.

From the Hartogs’s theorem and the theory of several complex variables, we
assume that

$\displaystyle
f(\alpha,x,y,a,z,\tilde{r},\tilde{s})=\sum_{k=0}^{\infty}A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})z^{k}.$
(2.2)

On one hand, substituting (2.2) into (2) yields

$(x-q^{-1}y)\sum_{k=0}^{\infty}(1-q^{2k})A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})z^{k}\\\
=\sum_{k=0}^{\infty}\Big{\\{}q^{\alpha-\tilde{s}+k(\tilde{r}-\tilde{s})}+q^{-\tilde{s}+k(1+\tilde{r}-\tilde{s})}(1-aq^{\alpha})-aq^{-\tilde{s}+k(2+\tilde{r}-\tilde{s})}\Big{\\}}\Big{\\{}A_{k}(\alpha,x,q^{-1}y,a,\tilde{r},\tilde{s})-A_{k}(\alpha,qx,y,a,\tilde{r},\tilde{s})\Big{\\}}z^{k+1},$
(2.3)

which is equal to

$(x-q^{-1}y)\sum_{k=0}^{\infty}(1-q^{2k})A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})z^{k}\\\
=\sum_{k=0}^{\infty}q^{-\tilde{s}+k(\tilde{r}-\tilde{s})}\Big{\\{}q^{\alpha}+q^{k}(1-aq^{\alpha})-aq^{2k}\Big{\\}}\Big{\\{}A_{k}(\alpha,x,q^{-1}y,a,\tilde{r},\tilde{s})-A_{k}(\alpha,qx,y,a,\tilde{r},\tilde{s})\Big{\\}}z^{k+1}.$
(2.4)

Equating coefficients of $z^{k},\,k\geq 1$ on both sides of equation (2.4), we
see that

$(x-q^{-1}y)(1-q^{k})(1+q^{k})A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})\\\
=q^{-\tilde{s}+(k-1)(\tilde{r}-\tilde{s})}(q^{\alpha}+q^{k-1})(1-aq^{k-1})\Big{[}A_{k-1}(\alpha,x,q^{-1}y,a,\tilde{r},\tilde{s})-A_{k-1}(\alpha,qx,y,a,\tilde{r},\tilde{s})\Big{]},$
(2.5)

or

$\displaystyle A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})$
$\displaystyle=q^{\tilde{r}(k-1)-\tilde{s}k}\frac{(q^{\alpha}+q^{k-1})(1-aq^{k-1})}{(1-q^{k})(1+q^{k})}\cdot\frac{A_{k-1}(\alpha,x,q^{-1}y,a,\tilde{r},\tilde{s})-A_{k-1}(\alpha,qx,y,a,\tilde{r},\tilde{s})}{x-q^{-1}y}$
$\displaystyle=q^{\alpha+\tilde{r}(k-1)-\tilde{s}k}\frac{(1+q^{-\alpha+k-1})(1-aq^{k-1})}{(1-q^{k})(1+q^{k})}\cdot
D_{xy}\\{A_{k-1}(\alpha,x,y,a,\tilde{r},\tilde{s})\\}.$

The iteration then reveals that:

$\displaystyle
A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})=q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}\cdot
D_{xy}^{k}\\{A_{0}(\alpha,x,y,a,\tilde{r},\tilde{s})\\}.$

Letting $\displaystyle
f(\alpha,x,y,a,0,\tilde{r},\tilde{s})=A_{0}(\alpha,x,y,a,\tilde{r},\tilde{s})=\sum_{n=0}^{\infty}\mu_{n}p_{n}(x,y)$
yields

$\displaystyle
A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})=q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}\cdot\sum_{n=0}^{\infty}\mu_{n}\frac{(q;q)_{n}}{(q;q)_{n-k}}p_{n-k}(x,y),$
(2.6)

and we have

$\displaystyle f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$
$\displaystyle=\sum_{k=0}^{\infty}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}\sum_{n=0}^{\infty}\mu_{n}\frac{(q;q)_{n}}{(q;q)_{n-k}}p_{n-k}(x,y)z^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\mu_{n}\sum_{k=0}^{n}\begin{bmatrix}n\\\
k\\\ \end{bmatrix}_{q}\begin{bmatrix}\alpha\\\ k\\\
\end{bmatrix}_{-q}q^{\tau(\tilde{r},\tilde{s})+({}^{k}_{2})}(a;q)_{k}p_{n-k}(x,y)z^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\mu_{n}\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a).$

On the other hand, if $f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$ can be expanded
in terms of $\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$, we verify
that $f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$ satisfies equation (2). The proof
of Theorem 2 is complete.

##  3\. Generating function of generalized $q$-polynomials

In this section, we give the generating function of generalized
$q$-polynomials by the method of $q$-difference equations.

######  Theorem 6.

Let $\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ be the polynomials
defined as in (1.10).

  * •

For $|xt|<1,$ and $\tilde{r},\,\tilde{s}\in\mathbb{Z}^{\ast}$, we have:

$\displaystyle\sum_{n=0}^{\infty}\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\frac{t^{n}}{(q;q)_{n}}=\frac{(yt;q)_{\infty}}{(xt;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}(zt)^{k}.$
(3.1)

  * •

For $\tilde{r}=\tilde{s}=0$ in (3.1) and $\max\\{|zt|,|xt|\\}<1,$ we recover
the Cauchy polynomials as follows:

$\displaystyle\sum_{n=0}^{\infty}\frac{p_{n}(x,y)\,t^{n}}{(q;q)_{n}}\sum_{k=0}^{n}{n\,\atopwithdelims[
]\,k\,}_{q}{\alpha\,\atopwithdelims[
]\,k\,}_{-q}\frac{(-1)^{k}q^{k(k-n)}(a;q)_{k}}{p_{k}(y,xq^{1-n})}z^{k}=\frac{(yt;q)_{\infty}}{(xt;q)_{\infty}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}-q^{-\alpha},a;\\\
\\\ -q;\end{array}q;ztq^{\alpha}\right].$ (3.5)

###### Proof of Theorem 6.

By the Weierstrass $M$-test, the series
$\displaystyle\sum_{n=0}^{\infty}M_{n}$ is convergent when
$\displaystyle\lim_{n\to+\infty}\left|\frac{M_{n+1}}{M_{n}}\right|<1$. We
check that both sides of (3.1) are convergent if $|xt|<1,$ that is

$\displaystyle\lim_{n\to\infty}\left|\frac{\tilde{L}_{n+1}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\;t^{n+1}/(q;q)_{n+1}}{\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\;t^{n}/(q;q)_{n}}\right|=|xt|<1,$
$\displaystyle\lim_{n\to\infty}\left|\frac{t^{n+1}\sum_{k=0}^{n}\begin{bmatrix}n+1\\\
k\\\ \end{bmatrix}_{q}\begin{bmatrix}\alpha\\\ k\\\
\end{bmatrix}_{-q}q^{\tau(\tilde{r},\tilde{s})+({}^{k}_{2})}(a;q)_{k}p_{n+1-k}(x,y)z^{k}/(q;q)_{n+1}}{t^{n}\sum_{k=0}^{n}\begin{bmatrix}n\\\
k\\\ \end{bmatrix}_{q}\begin{bmatrix}\alpha\\\ k\\\
\end{bmatrix}_{-q}q^{\tau(\tilde{r},\tilde{s})+({}^{k}_{2})}(a;q)_{k}p_{n-k}(x,y)z^{k}/(q;q)_{n}}\right|=|xt|<1.$

Denoting by $f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$ the right-hand side of
equation (3.1), it can equivalently be written by

$\displaystyle f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}z^{k}\cdot\frac{t^{k}\;(yt;q)_{\infty}}{(xt;q)_{\infty}}$
(3.6)
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}z^{k}D_{xy}^{k}\left\\{\frac{(yt;q)_{\infty}}{(xt;q)_{\infty}}\right\\}.$
(3.7)

Letting $\displaystyle
f(\alpha,x,y,a,z,\tilde{r},\tilde{s})=\sum_{k=0}^{\infty}A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})z^{k}$
and

$\displaystyle
A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})=q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}D_{xy}^{k}\left\\{\frac{(yt;q)_{\infty}}{(xt;q)_{\infty}}\right\\},$
(3.8)

we obtain

$\displaystyle
A_{0}(\alpha,x,y,a,\tilde{r},\tilde{s})=\frac{(yt;q)_{\infty}}{(xt;q)_{\infty}},$
(3.9)

and
$f(\alpha,x,y,a,0,\tilde{r},\tilde{s})=A_{0}(\alpha,x,y,a,\tilde{r},\tilde{s})$.
Taking (3.9) into (3.8), we get:

$\displaystyle
A_{k}(\alpha,x,y,a,\tilde{r},\tilde{s})=q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}\cdot
D_{xy}^{k}\\{A_{0}(\alpha,x,y,a,\tilde{r},\tilde{s})\\}.$ (3.10)

Through above identities, $f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$ satisfies
the equation (2). So, we have

$\displaystyle f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$
$\displaystyle=\sum_{k=0}^{n}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}z^{k}\sum_{n=0}^{\infty}\frac{p_{n}(x,y)\;t^{n+k}}{(q;q)_{n}}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}z^{k}\frac{p_{n-k}(x,y)\;t^{n}}{(q;q)_{n-k}}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{t^{n}}{(q;q)_{n}}\sum_{k=0}^{n}\begin{bmatrix}n\\\
k\\\ \end{bmatrix}_{q}\begin{bmatrix}\alpha\\\ k\\\
\end{bmatrix}_{-q}q^{\tau(\tilde{r},\tilde{s})+({}^{k}_{2})}(a;q)_{k}p_{n-k}(x,y)z^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\frac{t^{n}}{(q;q)_{n}}\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a),$

which is the left-hand side of (3.1). The proof is complete.

######  Remark 7.

Setting $y=0$, in (3.1), we get the concluding remarks of [14]:

$\displaystyle\sum_{n=0}^{\infty}L_{\tilde{r},\tilde{s}}(\alpha,x,z,a)\frac{t^{n}}{(q;q)_{n}}=\frac{1}{(xt;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}(zt)^{k},\;\;\;|xt|<1.$
(3.11)

For $\alpha\to\infty,\,\tilde{r}=\tilde{s}=0,\,x\to y,\,y\to x,\,z=-z$ and
$a=-q$, in (3.1), we get the concluding remarks of [1]:

$\displaystyle\sum_{n=0}^{\infty}F_{n}(x,y,z;q)\frac{(-1)^{n}q^{({}^{n}_{2})}\,t^{n}}{(q;q)_{n}}=\frac{(xt,zt;q)_{\infty}}{(yt;q)_{\infty}},\;\;\;|yt|<1.$
(3.12)

##  4\. Rogers formula for generalized $q$-polynomials

In this section, we give and prove the Rogers formula for generalized
$q$-polynomials by using the $q$-difference equations, so that we derive
Rogers formula for the trivariate $q$-polynomials:

Chen and Liu [7] have defined the $q$-exponential operator as follows (see
[18]):

$\displaystyle T(bD_{a})=\sum_{n=0}^{\infty}\frac{(bD_{a})^{n}}{(q;q)_{n}},$
(4.1)

where the usual $q$-differential operator, or $q$-derivative, is defined by

$D_{a}f(a)=\frac{f(a)-f(qa)}{a}.$ (4.2)

The Leibniz rule for $D_{a}$ is the following identity, which is a variation
of $q$-binomial theorem [19]:

$\displaystyle
D_{a}^{n}\\{f(a)g(a)\\}=\sum_{k=0}^{n}q^{k(k-n)}\begin{bmatrix}n\\\ k\\\
\end{bmatrix}_{q}D_{a}^{k}\\{f(a)\\}D_{a}^{n-k}\Bigl{\\{}g\bigl{(}aq^{k}\bigr{)}\Bigr{\\}},$
(4.3)

where $D_{q}^{0}$ is understood as the identity. The following property of
$D_{a}$ is straightforward, but important.

######  Lemma 8.

$D_{a}^{n}\left\\{\frac{(as;q)_{\infty}}{(a\omega;q)_{\infty}}\right\\}=\omega^{n}\;\frac{(s/\omega;q)_{n}}{(as;q)_{n}}\;\frac{(as;q)_{\infty}}{(a\omega;q)_{\infty}}.$
(4.4)

######  Lemma 9.

For $k\in\mathbb{N}_{0}$ and $\max\\{|x\omega|,|xt|\\}<1$, we have:

$\displaystyle
T(tD_{\omega})\left\\{\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\omega^{k}\right\\}=\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\omega^{k}\sum_{j=0}^{k}\frac{(-1)^{j}q^{kj-({}^{j}_{2})}(q^{-k},x\omega;q)_{j}\,(t/\omega)^{j}}{(y\omega,q;q)_{j}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}y/x,0;\\\
\\\ y\omega q^{j};\end{array}q;xt\right].$ (4.8)

###### Proof.

By means of the Leibniz rule (4.3), the left-hand side of (4.8) equals

$\displaystyle\sum_{n=0}^{\infty}\frac{t^{n}}{(q;q)_{n}}\sum_{j=0}^{n}q^{j(j-n)}\begin{bmatrix}n\\\
j\\\
\end{bmatrix}_{q}D_{\omega}^{j}\\{\omega^{k}\\}D_{\omega}^{n-j}\left\\{\frac{(y\omega
q^{j};q)_{\infty}}{(x\omega q^{j};q)_{\infty}}\right\\}$
$\displaystyle\qquad\qquad\qquad=\sum_{n=0}^{\infty}\sum_{j=0}^{n}\frac{q^{j(j-n)}\;t^{n}}{(q;q)_{j}(q;q)_{n-j}}\frac{(q;q)_{k}}{(q;q)_{k-j}}\omega^{k-j}D_{\omega}^{n-j}\left\\{\frac{(y\omega
q^{j};q)_{\infty}}{(x\omega q^{j};q)_{\infty}}\right\\}$
$\displaystyle\qquad\qquad\qquad=\sum_{n=0}^{\infty}\sum_{j=0}^{\infty}\frac{q^{-jn}\;t^{n+j}}{(q;q)_{j}(q;q)_{n}}\frac{(q;q)_{k}}{(q;q)_{k-j}}\omega^{k-j}D_{\omega}^{n}\left\\{\frac{(y\omega
q^{j};q)_{\infty}}{(x\omega q^{j};q)_{\infty}}\right\\}\quad(\mbox{by
}(\ref{aberll}))$
$\displaystyle\qquad\qquad\qquad=\sum_{n=0}^{\infty}\sum_{j=0}^{\infty}\frac{q^{-jn}\;t^{n+j}}{(q;q)_{j}(q;q)_{n}}\frac{(q;q)_{k}}{(q;q)_{k-j}}\omega^{k-j}(xq^{j})^{n}\frac{(y/x;q)_{n}(y\omega
q^{j};q)_{\infty}}{(y\omega q^{j};q)_{n}(x\omega q^{j};q)_{\infty}}$
$\displaystyle\qquad\qquad\qquad=\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\omega^{k}\sum_{j=0}^{\infty}\frac{(x\omega;q)_{j}\,(t/\omega)^{j}}{(y\omega,q;q)_{j}}\frac{(q;q)_{k}}{(q;q)_{k-j}}\sum_{n=0}^{\infty}\frac{(y/x;q)_{n}(xt)^{n}}{(y\omega
q^{j},q;q)_{n}}$
$\displaystyle\qquad\qquad\qquad=\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\omega^{k}\sum_{j=0}^{k}\frac{(-1)^{j}q^{kj-({}^{j}_{2})}(q^{-k},x\omega;q)_{j}\,(t/\omega)^{j}}{(y\omega,q;q)_{j}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}y/x,0;\\\
\\\ y\omega q^{j};\end{array}q;xt\right],$

which is the right-hand side of (4.8).

The generalized Rogers-Szegö polynomials [9, 20] are defined as

$\displaystyle r_{n}(x,y)=\sum_{k=0}^{n}\begin{bmatrix}n\\\ k\\\
\end{bmatrix}_{q}x^{k}y^{n-k},$ (4.9)

where [20]

$r_{n}(x,y)=T(xD_{y})\\{y^{n}\\}.$ (4.10)

Now, we are in position to give and prove the following Rogers formula for
generalized $q$-polynomials by using the $q$-difference equations.

######  Theorem 10.

Let $\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)$ be the polynomials
defined as in (1.10).

  * •

For $|\omega x|<1$, we have:

$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\tilde{L}_{n+m}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\frac{t^{n}}{(q;q)_{n}}\frac{\omega^{m}}{(q;q)_{m}}\\\
=\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\sum_{k=0}^{\infty}\sum_{j=0}^{k}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}(\omega
z)^{k}}{(-q;q)_{k}(q;q)_{k-j}}\frac{(x\omega;q)_{j}(t/\omega)^{j}}{(y\omega,q;q)_{j}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}y/x,0;\\\
\\\ y\omega q^{j};\end{array}q;xt\right].$ (4.11)

  * •

For $\max\\{|\omega x|,|xt|\\}<1$, and $\tilde{r}=\tilde{s}=0,$ we have:

$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\tilde{L}_{n+m}^{(0,0)}(\alpha,x,y,z,a)\frac{t^{n}}{(q;q)_{n}}\frac{\omega^{m}}{(q;q)_{m}}\\\
=\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\sum_{k=0}^{\infty}\sum_{j=0}^{k}q^{k\alpha}\frac{(-q^{-\alpha},a;q)_{k}(\omega
z)^{k}}{(-q;q)_{k}(q;q)_{k-j}}\frac{(x\omega;q)_{j}(t/\omega)^{j}}{(y\omega,q;q)_{j}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}y/x,0;\\\
\\\ y\omega q^{j};\end{array}q;xt\right].$ (4.12)

###### Proof of Theorem 10.

The right-hand side of equation (4.11) can equivalently be written as:

$\displaystyle f(\alpha,x,y,a,z,\tilde{r},\tilde{s})$
$\displaystyle=\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}(\omega
z)^{k}}{(-q;q)_{k}(q;q)_{k-j}}\frac{(x\omega;q)_{j}\,(t/\omega)^{j}}{(y\omega,q;q)_{j}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}y/x,0;\\\
\\\ y\omega q^{j};\end{array}q;xt\right]$
$\displaystyle=\sum_{k=0}^{\infty}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}z^{k}}{(q^{2};q^{2})_{k}}\sum_{j=0}^{k}\frac{(y\omega
q^{j};q)_{\infty}(q;q)_{k}\omega^{k-j}t^{j}}{(x\omega
q^{j};q)_{\infty}(q;q)_{j}(q;q)_{k-j}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}y/x,0;\\\
\\\ y\omega q^{j};\end{array}q;xt\right].$
$\displaystyle=\sum_{k=0}^{\infty}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}z^{k}T(tD_{\omega})\left\\{\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\omega^{k}\right\\}$
$\displaystyle=T(tD_{\omega})\left\\{\frac{(y\omega;q)_{\infty}}{(x\omega;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}(\omega
z)^{k}\right\\}\;\,(\mbox{by using }(\ref{thmc3_1}))$
$\displaystyle=T(tD_{\omega})\left\\{\sum_{m=0}^{\infty}\tilde{L}_{m}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\frac{\omega^{m}}{(q;q)_{m}}\right\\}$
$\displaystyle=\sum_{m=0}^{\infty}\tilde{L}_{m}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\frac{1}{(q;q)_{m}}T(tD_{\omega})\\{\omega^{m}\\}\quad(\mbox{by
}(\ref{RSpo}))$
$\displaystyle=\sum_{m=0}^{\infty}\tilde{L}_{m}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\frac{r_{m}(t,\omega)}{(q;q)_{m}}$
$\displaystyle=\sum_{m=0}^{\infty}\tilde{L}_{m}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\frac{1}{(q;q)_{m}}\sum_{n=0}^{m}\begin{bmatrix}m\\\
n\\\ \end{bmatrix}_{q}t^{n}\omega^{m-n}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{m=n}^{\infty}\tilde{L}_{m}^{(\tilde{r},\tilde{s})}(\alpha,x,y,z,a)\frac{t^{n}}{(q;q)_{n}}\frac{\omega^{m-n}}{(q;q)_{m-n}}.$

By setting $m\to m+n$, we get the left-hand side of (4.11). This completes the
proof.

As a special case of Theorem 10, if we take
$\alpha\to\infty,\,\tilde{r}=\tilde{s}=0,\,x\to y,\,y\to x,\,z=-z$ and $a=-q$,
in (4.11), we obtain the following corollary:

######  Corollary 11 ([1, Theorem 3.1]).

For $\max\\{|yt|,|\omega y|\\}<1$, we have:

$\displaystyle\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}F_{n+m}(x,y,z;q)(-1)^{n+m}q^{({}^{n+m}_{\,\;\;\;2})}\frac{t^{n}}{(q;q)_{n}}\frac{\omega^{m}}{(q;q)_{m}}$
(4.13)
$\displaystyle\qquad\quad\quad\qquad\qquad=\frac{(x\omega,z\omega;q)_{\infty}}{(y\omega;q)_{\infty}}\sum_{j=0}^{\infty}\frac{(-1)^{j}q^{({}^{j}_{2})}(y\omega;q)_{j}(zt)^{j}}{(x\omega,z\omega,q;q)_{j}}{}_{2}\Phi_{1}\left[\begin{array}[]{r}x/y,0;\\\
\\\ x\omega q^{j};\end{array}q;yt\right].$ (4.17)

##  5\. Mixed generating function for generalized $q$-polynomials

The Hahn polynomials [12, 13] (or Al-Salam and Carlitz polynomials [2, 6]) are
defined as

$\phi_{n}^{(\sigma)}(x|q)=\sum_{k=0}^{n}\begin{bmatrix}n\\\ k\\\
\end{bmatrix}_{q}(\sigma;q)_{k}\,x^{k}.$ (5.1)

While Cao [6] used the technique of exponential operator decomposition,
Srivastava and Agarwal [23] adopted the method of transformation theory to
deduce the following results (For more information, please refer to [12, 13,
2, 23, 4, 5, 6]:

######  Lemma 12 ([23, Eq. (3.20)]).

For $\max\\{|t|,|xt|\\}<1$, we have:

$\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)(\lambda;q)_{n}\frac{t^{n}}{(q;q)_{n}}\\\
=\frac{(\lambda
t;q)_{\infty}}{(t;q)_{\infty}}{}_{2}\Phi_{1}\left[\begin{array}[]{rr}\lambda,\sigma;\\\
\\\ \lambda t;\end{array}\,q;xt\right].$ (5.2)

In Theorem 13 below, we give and prove a mixed generating function for
generalized $q$-polynomials by applying the $q$-difference equations.

######  Theorem 13.

Let $\tilde{r},\,\tilde{s}\in\mathbb{Z}$. For $|ut|<1,$ we have:

$\displaystyle\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,u,v,z,a)\frac{\,t^{n}}{(q;q)_{n}}$
(5.3)
$\displaystyle\qquad\qquad=\frac{(vt;q)_{\infty}}{(ut;q)_{\infty}}\sum_{m=0}^{\infty}\sum_{k=0}^{\infty}\sum_{j=0}^{m}\frac{(\sigma;q)_{m}\,x^{m}}{(q;q)_{m}}\frac{(q^{-m},ut;q)_{j}q^{j}}{(vt,q;q)_{j}}\frac{(-q^{-\alpha};q)_{k}(a;q)_{k}\,(tzq^{j})^{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}.$
(5.4)

In the proof of Theorem 13, the following $q$-Chu-Vandermonde formula will be
needed:

######  Lemma 14 ($q$-Chu-Vandermonde [10, Eq. (II.6)]).

${}_{2}\Phi_{1}\left[\begin{array}[]{rr}q^{-n},a;\\\ \\\
c;\end{array}\,q;q\right]=\frac{(c/a;q)_{n}}{(c;q)_{n}}a^{n}.$ (5.5)

###### Proof of Theorem 13.

Eq. (5.3) can equivalently be written as follows:

$\displaystyle\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,u,v,z,a)\frac{\,t^{n}}{(q;q)_{n}}$
(5.6)
$\displaystyle\qquad\qquad=\sum_{m=0}^{\infty}\frac{(\sigma;q)_{m}\,x^{m}}{(q;q)_{m}}\sum_{j=0}^{m}\frac{(q^{-m};q)_{j}q^{j}}{(q;q)_{j}}\sum_{k=0}^{\infty}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha};q)_{k}(a;q)_{k}\,z^{k}}{(q^{2};q^{2})_{k}}D_{uv}^{k}\left\\{\frac{(vtq^{j};q)_{\infty}}{(utq^{j};q)_{\infty}}\right\\}.$
(5.7)

If we use $g(\alpha,u,v,a,z,\tilde{r},\tilde{s})$ to denote the right-hand
side of (5.6), it is easy to see that $g(\alpha,u,v,a,z,\tilde{r},\tilde{s})$
satisfies (2). Letting $\displaystyle
g(\alpha,u,v,a,z,\tilde{r},\tilde{s})=\sum_{k=0}^{\infty}B_{k}(\alpha,u,v,a,\tilde{r},\tilde{s})z^{k}$
and

$\displaystyle
B_{k}(\alpha,u,v,a,\tilde{r},\tilde{s})=q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha};q)_{k}(a;q)_{k}}{(q^{2};q^{2})_{k}}D_{uv}^{k}\left\\{\sum_{m=0}^{\infty}\frac{(\sigma;q)_{m}\,x^{m}}{(q;q)_{m}}\sum_{j=0}^{m}\frac{(q^{-m};q)_{j}q^{j}}{(q;q)_{j}}\frac{(vtq^{j};q)_{\infty}}{(utq^{j};q)_{\infty}}\right\\},$
(5.8)

we obtain

$\displaystyle B_{0}(\alpha,u,v,a,\tilde{r},\tilde{s})$
$\displaystyle=\frac{(vt;q)_{\infty}}{(ut;q)_{\infty}}\sum_{m=0}^{\infty}\frac{(\sigma;q)_{m}\,x^{m}}{(q;q)_{m}}\sum_{j=0}^{m}\frac{(q^{-m},ut;q)_{j}q^{j}}{(vt,q;q)_{j}}$
(5.9)
$\displaystyle=\frac{(vt;q)_{\infty}}{(ut;q)_{\infty}}\sum_{m=0}^{\infty}\frac{(\sigma;q)_{m}\,x^{m}}{(q;q)_{m}}{}_{2}\Phi_{1}\left[\begin{array}[]{rr}q^{-m},ut;\\\
\\\ vt;\end{array}\,q;q\right]\quad(\mbox{by }(\ref{male}))$ (5.13)
$\displaystyle=\frac{(vt;q)_{\infty}}{(ut;q)_{\infty}}\sum_{m=0}^{\infty}\frac{(\sigma;q)_{m}\,x^{m}}{(q;q)_{m}}.\frac{(v/u;q)_{m}(ut)^{m}}{(vt;q)_{m}}$
(5.14)
$\displaystyle=\frac{(vt;q)_{\infty}}{(ut;q)_{\infty}}{}_{2}\Phi_{1}\left[\begin{array}[]{rr}v/u,\sigma;\\\
\\\ vt;\end{array}\,q;uxt\right]$ (5.18)
$\displaystyle=\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)\frac{p_{n}(u,v)\,t^{n}}{(q;q)_{n}},$
(5.19)

and
$g(\alpha,u,v,a,0,\tilde{r},\tilde{s})=B_{0}(\alpha,u,v,a,\tilde{r},\tilde{s})$.
Taking equation (5.19) into (5.8), we get:

$\displaystyle
B_{k}(\alpha,u,v,a,\tilde{r},\tilde{s})=q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}\cdot
D_{uv}^{k}\\{B_{0}(\alpha,u,v,a,\tilde{r},\tilde{s})\\}.$ (5.20)

Through above identities, $g(\alpha,u,v,a,z,\tilde{r},\tilde{s})$ satisfies
the equation (2). So, we have

$\displaystyle g(\alpha,u,v,a,z,\tilde{r},\tilde{s})$
$\displaystyle=\sum_{k=0}^{n}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}z^{k}D_{uv}^{k}\left\\{\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)\frac{p_{n}(u,v)\,t^{n}}{(q;q)_{n}}\right\\}$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{(-q^{-\alpha},a;q)_{k}}{(q^{2};q^{2})_{k}}q^{k\alpha+\tilde{r}({}^{k}_{2})-\tilde{s}({}^{k+1}_{\;\,\;2})}z^{k}\phi_{n}^{(\sigma)}(x|q)\frac{p_{n-k}(u,v)\;t^{n}}{(q;q)_{n-k}}$
$\displaystyle=\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)\frac{t^{n}}{(q;q)_{n}}\sum_{k=0}^{n}\begin{bmatrix}n\\\
k\\\ \end{bmatrix}_{q}\begin{bmatrix}\alpha\\\ k\\\
\end{bmatrix}_{-q}q^{\tau(\tilde{r},\tilde{s})+({}^{k}_{2})}(a;q)_{k}p_{n-k}(u,v)z^{k}$
$\displaystyle=\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)\tilde{L}_{n}^{(\tilde{r},\tilde{s})}(\alpha,u,v,z,a)\frac{t^{n}}{(q;q)_{n}},$

which is the left hand side of (5.3). The proof is complete.

As a special case of Theorem 13, if we take $\alpha\to\infty,\;a=-q,\,u\to
v,\,v\to u,\,z=-z$ and $\tilde{r}=\tilde{s}=0$, we have the following
corollary:

######  Corollary 15 (Mixed generating function for trivariate $q$-polynomials
$F_{n}(x,y,z;q)$).

For $\max\\{|vt|,|x|\\}<1,$ we have:

$\displaystyle\sum_{n=0}^{\infty}\phi_{n}^{(\sigma)}(x|q)F_{n}(u,v,z;q)\frac{(-1)^{n}q^{({}^{n}_{2})}\,t^{n}}{(q;q)_{n}}=\frac{(\sigma
x,ut,zt;q)_{\infty}}{(vt,x;q)_{\infty}}{}_{4}\Phi_{3}\left[\begin{array}[]{rr}\sigma,vt,0,0;\\\
\\\ ut,zt,q/x;\end{array}\,q;q\right].$ (5.24)

##  6\. Concluding remarks

In this paper, by using the method of $q$-difference equations, we have
systematically deduced several types of generating functions for certain
$q$-polynomials. In fact, $q$-Laguerre polynomials and Cigler’s polynomials
have double $q$-binomial coefficients. Computing their generating functions
reveals to be too difficult. We have noticed that the $q$-difference equations
are important tools to calculate generating functions for $q$-polynomials. It
is necessary to construct $q$-difference equations satisfied by the general
$q$-polynomials with double $q$-binomial coefficients. We have then focused on
the expansion of a function of many variables and on some $q$-polynomials.
Therefore, we have searched for the generalized $q$-difference equations for
general $q$-polynomials with double $q$-binomial coefficients. We believe that
this work will be a motivation to study other $q$-polynomials and their
applications.

## Acknowledgments

The authors are grateful to the referees and editor for their useful comments
and suggestions to improve the paper. This work was supported by the Zhejiang
Provincial Natural Science Foundation of China (No. LY21A010019). The ICMPA-
UNESCO Chair is in partnership with the Association pour la Promotion
Scientifique de l’Afrique (APSA), France, and Daniel Iagolnitzer Foundation
(DIF), France, supporting the development of mathematical physics in Africa.

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# Learning Continuous Mesh Representation with
Spherical Implicit Surface

Zhongpai Gao1,2
1 MoE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong
University, Shanghai, China 2 United Imaging Intelligence, Cambridge MA, USA
Copyright notice: 979-8-3503-4544-5/23/$31.00 ©2023 IEEE. This work was
supported by the National Natural Science Foundation of China (61901259) and
China Postdoctoral Science Foundation (BX2019208).

###### Abstract

As the most common representation for 3D shapes, mesh is often stored
discretely with arrays of vertices and faces. However, 3D shapes in the real
world are presented continuously. In this paper, we propose to learn a
continuous representation for meshes with fixed topology, a common and
practical setting in many faces-, hand-, and body-related applications. First,
we split the template into multiple closed manifold genus-0 meshes so that
each genus-0 mesh can be parameterized onto the unit sphere. Then we learn
spherical implicit surface (SIS), which takes a spherical coordinate and a
global feature or a set of local features around the coordinate as inputs,
predicting the vertex corresponding to the coordinate as an output. Since the
spherical coordinates are continuous, SIS can depict a mesh in an arbitrary
resolution. SIS representation builds a bridge between discrete and continuous
representation in 3D shapes. Specifically, we train SIS networks in a self-
supervised manner for two tasks: a reconstruction task and a super-resolution
task. Experiments show that our SIS representation is comparable with state-
of-the-art methods that are specifically designed for meshes with a fixed
resolution and significantly outperforms methods that work in arbitrary
resolutions.

##  I INTRODUCTION

3D shapes in the real world are continuous. While, in the digital world, we
usually capture, store, and process 3D shapes in a discrete way. A common
representation of 3D shapes is triangulated mesh that structures a 3D shape as
arrays of vertices and faces. The precision of mesh representation for 3D
shapes is controlled by resolution (i.e., number of vertices). The vertex-
based mesh representation has been widely applied in many computer vision and
computer graphics applications, e.g., 3D reconstruction [12, 29, 11], shape
correspondence [15], virtual avatar [5], gesture synthesis [23], etc. However,
the vertex-based mesh representation is difficult for applications that
require various mesh resolutions. In this paper, we propose a continuous
representation for meshes. By modeling a mesh as a function defined in a
continuous domain, we can process the mesh in an arbitrary resolution as
needed.

Closed manifold genus-0 meshes are topologically equivalent to a sphere, hence
this is the natural and continuous parameter domain for them, called spherical
parameterization [14]. Specifically, spherical conformal parameterization [3,
7] that preserves the angle and hence the local geometry of the surface is the
most important type of parameterization since the angle structure plays an
important role in the computation of texture mapping, remeshing, and many
other applications. Thus, spherical conformal parameterization provides a one-
to-one correspondence between meshes and a sphere such that the spherical
coordinate can be considered as the canonical coordinate in a continuous
domain for 3D shapes. Inspired by the continuous image representation [6] that
models an image as an implicit function of the continuous 2D coordinates, we
model a mesh as an implicit function of the continuous spherical coordinates.
The implicit function can be parameterized by a deep neural network, e.g.,
multilayer perceptions (MLP) to map each coordinate to the corresponding
surface position of the 3D shape. Note that, for a mesh that is not closed
manifold genus-0, we always can split the mesh into multiple closed manifold
genus-0 meshes with the help of filling holes if necessary.

This paper proposes spherical implicit surface (SIS) for representing mesh in
a continuous manner. SIS can represent a mesh with an arbitrary topology.
While, in this paper, we mainly focus on the SIS representation for a group of
meshes with the same topology, e.g., faces, bodies, and hands. To share
knowledge across samples instead of fitting individual implicit function for
each mesh, we use an encoder to predict a global feature for each mesh. Then
the implicit function is shared by all the meshes while it is conditioned upon
the global feature in addition to the spherical coordinates as inputs. At
last, the implicit function predicts the 3D position at the given spherical
coordinate as the output. Furthermore, instead of using one global feature to
encode the whole mesh, we represent a mesh by a set of local features
distributed in spatial dimensions (i.e., 3D shape surface). Given a spherical
coordinate, the implicit function takes the coordinate information and queries
the local features around the coordinate as inputs, then predicts the 3D
position at the given coordinate as the output. Either given the global
feature or a set of local features of a mesh, the SIS representation can
present the mesh in an arbitrary resolution since the spherical coordinates
are continuous.

To learn SIS continuous representation from the global feature of a mesh, we
train a mesh encoder and an SIS decoder via a reconstruction task in a self-
supervised manner. The mesh encoder is built by a convolutional operation
named LSA-Conv [10] to extract the global feature of a mesh. To learn SIS
continuous representation from a set of local features of a mesh, we train an
SIS encoder, a feature fusion module, and an SIS decoder via a super-
resolution task in a self-supervised manner. The SIS encoder takes the vertex
information in addition to the spherical coordinate as inputs and predicts the
corresponding deep feature as the output. The local feature of a spherical
coordinate in a higher resolution is assembled by the feature fusion module
which makes use of barycentric coordinates for interpolation. SIS builds a
bridge between the discrete and continuous representation in mesh and can
naturally exploit the information provided in different resolutions. The SIS
representation can present a mesh in an arbitrary resolution, thus it can be
trained without resizing ground-truths and achieves better results than
methods designed for a certain resolution. We evaluate our approach on the
reconstruction and super-resolution task in two 3D shape datasets: human faces
(COMA [27]) and human bodies (DFAUST [4]).

The contributions of this paper are summarized in below:

1) Taking advantage of that genus-0 meshes are topologically equivalent to a
sphere, we use spherical conformal parameterization to map meshes to a sphere
as the continuous canonical coordinate for 3D shapes. Then, we introduce a new
continuous mesh representation by modeling a mesh as an implicit function of
the spherical coordinates, called spherical implicit surface (SIS).

2) We show how this continuous representation can be used for reconstructing
meshes. In addition to the spherical coordinates, the SIS representation
either takes the global feature or a set of local features of a mesh as inputs
to present the mesh in a continuous manner. For the input of local features,
we introduce a feature fusion module that makes use of barycentric coordinates
for interpolation to bridge between the discrete and continuous domains.

3) Extensive experiments on COMA [27] and DFAUST [4] datasets show that our
approach is able to generate high-quality meshes and demonstrate that it
compares favorably to state-of-the-art methods designed for discrete domains
and outperforms methods designed for continuous domains.

##  II RELATED WORK

Figure 1: Spherical coordinates for a mesh. The facial mesh $M$ in (a) has
multiple components and are not a closed manifold genus-0 mesh. It can be
split into multiple genus-0 meshes: the left eye $M_{g2}$, the right eye
$M_{g3}$, and the rest part $M_{g1}$ in (b). Each genus-0 mesh $M_{gi}$ can be
mapped to the unit sphere $S^{2}$ by applying spherical conformal
parameterization $f_{i}:S^{2}\rightarrow M_{gi}$ from the unit sphere $S^{2}$
to the genus-0 mesh $M_{gi}$. The unit sphere $S^{2}$ can be parameterized by
the normalized inclination angle $\hat{\theta}$ and azimuth angle
$\hat{\varphi}$ such that we create a one-to-one correspondence
$f_{i}:(\hat{\theta},\hat{\varphi})\rightarrow{\bm{V}}_{i}$ from the spherical
coordinates to the vertices of the genus-0 mesh. The colors in (c) and (d)
represent the corresponding vertex position on the mesh.

###  II-A Discrete representations for 3D shapes

Discrete representations for learning-based 3D tasks can be mainly categorized
as: voxel-based, point-based, and mesh-based. Voxel presentation is a
straightforward generalization of pixels to the 3D cases and has been used for
discriminative [19, 26] and generative [8, 13] 3D tasks. However, voxel
representations require memory that grows cubically with resolution. Point
clouds [1, 9] and meshes [17, 11] have been introduced as alternative
representations for deep learning. However, point clouds lack the connectivity
structure of the 3D shapes and usually require additional post-processing to
generate the final 3D shape. Moreover, all the discrete representations are
limited to the resolution (i.e., number of points/vertices) that a model can
produce.

In contrast to the discrete representations, our approach leads to continuous
surface of 3D shapes. Using deep learning, our approach obtains a more
expressive representation that can naturally be integrated into existing 3D
shape generation pipelines [11, 15].

###  II-B Implicit Representations

Implicit representations are continuous and differentiable functions that map
coordinates to signal [28], e.g., images and 3D shapes, and are parameterized
as multilayer perceptions (MLP). For images, [6] proposed local implicit image
function (LIIF) that takes an image coordinate and the 2D deep features around
the coordinate as inputs to predict the RGB value at a given coordinate so
that the learned representation can present an image in an arbitrary
resolution.

For 3D shapes, recent work has investigated implicit representations of
continuous 3D shapes that map $xyz$ coordinates to a signed distance function
(SDF) [24] or to an occupancy field [20, 25] or to a neural radiance field
(NeRF) [21]. SDF represents a 3D shape’s surface by a continuous volumetric
field — the distance of a point to the surface boundary and the sign indicates
whether the region is inside or outside of the shape, thus it implicitly
encodes a shape’s boundary as the zero-level-set of the learned function.
Occupancy field is a special case of SDF and only considers the ‘sign’ of SDF
values to classify 3D points as inside or outside of a 3D shape. NeRF
represents a scene by the volume density and view-dependent emitted radiance
of a point and can produce high-fidelity appearance to render photorealistic
novel views of complex 3D scenes. Another continuous representation for 3D
shapes was introduced by [15], called template deformation (TDeform) that uses
an MLP to regress the point-wise deformation of 3D shapes from the template in
any resolution.

However, for the implicit representations of SDF [24], occupancy field [20],
and NeRF [21], the coordinates are defined as $xyz$ positions in a volumetric
space, which requires large amounts of samples from the volumetric space for
training and needs an isosurface extraction algorithm for inference to extract
the surface from a trained model. Compared to those implicit representations
of 3D shapes, our SIS representation directly works on the surface and is more
efficient for both training and inference. Similar to the one-to-one mapping
from image coordinates to images, our SIS representation has a one-to-one
mapping from the spherical coordinates to the surface of 3D shapes such that
we only need to train on the samples of 3D shape vertices and infer a 3D shape
simply by inputting the spherical coordinates.

Even though TDeform [15] that defines the coordinates as the template vertices
creates a one-to-one mapping from the template to the surface of a 3D shape,
the coordinates are $xyz$ positions in a volumetric space and most of the
coordinates (except for the template vertices) do not have the corresponding
labels, making the network difficult to be trained (not bijective). In
contrast, our spherical coordinates are continuous and corresponding to the
surface of 3D shapes everywhere (bijective). Thus, our approach is an
efficient and effective continuous representation for 3D shapes.

##  III SPHERICAL IMPLICIT SURFACE

In this section, we introduce spherical implicit surface (SIS) — our
continuous representation for meshes. First, we apply spherical conformal
parameterization to have a one-to-one mapping from a mesh to the unit sphere
so that the spherical coordinate can be used as the continuous canonical
coordinate for the mesh. Then, we describe how we can learn an SIS network
that takes a global feature or a set of local features in addition to the
spherical coordinates for 3D shape generations. At last, we introduce the loss
function used to train our models.

###  III-A Spherical Coordinate

Figure 2: Learning to generate spherical implicit surface (SIS) representation
for meshes. (a) The SIS representation inputs the spherical coordinates and
outputs the mesh vertices and is fitted to an individual mesh. (b) The SIS
representation is conditioned with a global feature that is extracted from a
mesh using a mesh encoder. (c) The SIS representation is conditioned with a
local feature for each spherical coordinate. We input the SIS encoder a
subsempled mesh whose topology is created with the help of spherical mapping
and output the deep feature for each input vertex. A feature fusion module is
introduced to ensemble the local feature for higher-resolution spherical
coordinates based on barycentric coordinates.

Inspired by images where RGB values and image coordinates (i.e., $xy$
coordinates) are one-to-one corresponding to each other, we seek for a one-to-
one mapping between a canonical coordinate and meshes — a bijectve function.

###### Theorem 1

The ‘uniformization theorem’ guarantees that there is a conformal map
$f:S^{2}\rightarrow M$ from the unit sphere $S^{2}$ to any genus-0 mesh $M$,
i.e., a smooth, nondegenerate, and globally injective map that preserves both
angles and orientation.

A mesh can be defined as $M=(\bm{\mathcal{V}},\bm{\mathcal{F}})$, where
$\bm{\mathcal{V}}=\\{{\bm{v}}_{1},\ldots,{\bm{v}}_{N}\\}$ is a set of $N$
vertices and
$\bm{\mathcal{F}}\subseteq\bm{\mathcal{V}}\times\bm{\mathcal{V}}\times\bm{\mathcal{V}}$
is a set of triangular faces. The conformal map $f:S^{2}\rightarrow M$ can be
achieved by applying spherical conformal parameterization [3, 7] on a genus-0
mesh. As shown in Figure 1, when a mesh is not a genus-0, we can always split
the mesh $M$ into multiple submeshes $\\{M_{1},\cdots,M_{K}\\}$, where
$M=\\{M_{1},\ldots,M_{K}\\}$. We may close the holes of the submeshes $M_{i}$
as necessary [2] so that each submesh is a closed manifold genus-0 mesh
$M_{gi}$, where $i\in\\{1,\ldots,K\\}$. The original submesh $M_{i}$ is a
subset of the genus-0 mesh $M_{gi}$, i.e., $M_{i}=M_{gi}[\mathcal{I}_{i}]$
where $\mathcal{I}_{i}$ is the vertex index of $M_{i}$ in $M_{gi}$. Thus, any
mesh can be formulated with one or multiple genus-0 meshes
$M=\\{M_{g1}[\mathcal{I}_{1}],\ldots,M_{gK}[\mathcal{I}_{K}]\\}$ where
$K\in\mathbb{N}$. Then each genus-0 submesh $M_{gi}$ can find a conformal map
$f_{i}:S^{2}\rightarrow M_{gi}$ from the unit sphere $S^{2}$ to the genus-0
submesh $M_{gi}$. For simplicity and without losing generality, we assume the
mesh $M$ is already a closed manifold genus-0 mesh where $K=1$ and
$\mathcal{I}=[1,\ldots,N]$.

The surface of the unit sphere $S^{2}$ can be parameterized by two numbers:
its polar (inclination) angle $\theta$ measured from a fixed zenith direction
and the azimuthal angle $\varphi$ of its orthogonal projection on a reference
plane that passes through the origin and is orthogonal to the zenith,
expressed as

$\displaystyle\theta$
$\displaystyle=\arctan\frac{\sqrt{x^{2}+y^{2}}}{z}\in[0,\pi],$
$\displaystyle\varphi$ $\displaystyle=\arctan\frac{y}{x}\in[-\pi,\pi],$ (1)
$\displaystyle\hat{\theta}$ $\displaystyle=\frac{\theta}{\pi}\in[0,1],$
$\displaystyle\hat{\varphi}$ $\displaystyle=\frac{\varphi}{2\pi}+0.5\in[0,1],$
(2)

where $\hat{\theta}$ and $\hat{\varphi}$ are the normalized inclination and
azimuthal angle of a point on the unit sphere. Thus, we create a one-to-one
correspondence $f:(\hat{\theta},\hat{\varphi})\rightarrow{\bm{V}}$ from the
spherical coordinates to the mesh vertices, i.e.,
${\bm{v}}_{j}=f(\hat{\theta}_{j},\hat{\varphi}_{j})$, where
$j\in\\{1,\ldots,N\\}$. As shown in Fig. 2a, the implicit function $f$ that is
parameterized as an MLP network can be trained in a supervised manner. The
trained implicit function $f$ is a continuous representation for the mesh,
called spherical implicit surface (SIS) representation. During the inference
phase, we can take $(\hat{\theta},\hat{\varphi})$ continuously to generate the
3D shape in a higher resolution.

Though an MLP networks are universal function approximations [16], directly
inputting the coordinates $(\hat{\theta},\hat{\varphi})$ performs poorly at
representing high-frequency variation in geometry and Fourier feature mapping
enables an MLP network to learn high-frequency functions [28]. Inspired by
NeRF [21], we encode the spherical coordinates as

$\leavevmode\resizebox{420.43271pt}{}{ $\xi(p)=(\sin(2^{0}\pi p),\cos(2^{0}\pi
p),\ldots,\sin(2^{L-1}\pi p),\cos(2^{L-1}\pi p))$},$ (3)

where $p=(\hat{\theta},\hat{\varphi})$ and $L=10$ in our experiments. Though
Fourier feature mapping $\xi(\cdot)$ has been used in NeRF, applying it on our
spherical coordinates is physically more meaningful than on the $xyz$
coordinates used in NeRF since the spherical coordinates
$(\hat{\theta},\hat{\varphi})$ are defined in angles as presented in Eq. (1)
and Fig. 1d are periodic, which is naturally suitable for Fourier feature
mapping.

###  III-B Condition with Global Feature

Instead of fitting the implicit function $f$ to an individual mesh $M$, we
propose an SIS representation that is shared by a group of meshes, which can
be achieved by conditioning an observation of that mesh on the input in
addition to the spherical coordinates. We train the model in a self-supervised
manner via a reconstruction task. The observation of a mesh can be considered
as a global feature ${\bm{z}}_{g}$ extracted by a mesh encoder, as shown in
Fig. 2b. During the inference phase, we can use the implicit function to
reconstruct a mesh in an arbitrary resolution given its global feature. Thus,
the implicit function (i.e., SIS decoder) can be expressed as

$\displaystyle{\bm{v}}=f({\bm{z}}_{g},\hat{\theta},\hat{\varphi}),$ (4)

where ${\bm{z}}_{g}=en_{g}(M)$ and $en_{g}(\cdot)$ is the mesh encoder built
by convolutional operations and LSA-Conv [10] is used in our experiments.

###  III-C Condition with Local Feature

To make the SIS representation more expressive, instead of using one global
feature to encode the whole mesh, we encode a mesh by a set of local features
distributed in spatial dimensions such that each of them stores information
about its local area. We train the model in a self-supervised manner via a
super-resolution task. Thus, the input is noisy sparse point cloud that is
randomly sampled from the mesh (step 1 in Fig. 2c). Based on the spherical
mapping, we can find the points on the sphere corresponding to the point cloud
(step 2 in Fig. 2c). Then, we can easily and consistently build a topology
connection for the corresponding points on the sphere, which is the same for
the point cloud, thus we build a subsampled mesh (i.e., a lower resolution
mesh) for the randomly sampled point cloud (step 3 in Fig. 2c).

For a subsampled mesh, the SIS encoder maps each vertex ${\bm{v}}_{i}$ to a
deep feature ${\bm{z}}_{li}$. Note that, the spherical coordinates of the
subsampled mesh can correspond to any point on the sphere since the SIS
encoder is a continuous representation. The SIS decoder is also a continuous
representation and may take spherical coordinates that are not provided in the
subsampled mesh, i.e., spherical coordinates in a higher resolution. Thus, the
SIS encoder cannot provide the deep feature for those higher-resolution
spherical coordinates. We propose a feature fusion module based on barycentric
coordinates to obtain the local feature given any spherical coordinate.

Given a pair of spherical coordinate $(\hat{\theta},\hat{\varphi})$, we first
find the triangular face that contains the spherical coordinate on the sphere
that has the same topology of the subsampled mesh. We denote the spherical
coordinates of the triangular vertices on the sphere as
$[(\hat{\theta}_{1},\hat{\varphi}_{1}),(\hat{\theta}_{2},\hat{\varphi}_{2})$,
$(\hat{\theta}_{3},\hat{\varphi}_{3})]$ and denote the triangular vertices of
the subsampled mesh as [${\bm{v}}_{1},{\bm{v}}_{2}$, ${\bm{v}}_{3}]$. The deep
features of the triangular vertices are
${\bm{z}}_{l1}=en_{l}({\bm{v}}_{1}),{\bm{z}}_{l2}=en_{l}({\bm{v}}_{2})$, and
${\bm{z}}_{l3}=en_{l}({\bm{v}}_{3})$, where $en_{l}(\cdot)$ is the SIS
encoder. We can calculate the barycentric coordinates for the spherical
coordinate $(\hat{\theta},\hat{\varphi})$ relative to the three triangular
vertices as $[\lambda_{1},\lambda_{2},\lambda_{3}]$ where
$\sum_{i=1}^{3}\lambda_{i}=1$. Thus, based on the barycentric coordinates, we
can obtain a coarse deep feature for the spherical coordinate
$(\hat{\theta},\hat{\varphi})$ as,

$\displaystyle\hat{{\bm{z}}}_{l}=\lambda_{1}{\bm{z}}_{l1}+\lambda_{2}{\bm{z}}_{l2}+\lambda_{3}{\bm{z}}_{l3}.$
(5)

The feature fusion module ensembles the local feature for the spherical
coordinate $(\hat{\theta},\hat{\varphi})$ as

$\displaystyle{\bm{z}}_{l}=\hat{{\bm{z}}}_{l}\oplus(\hat{{\bm{z}}}_{l}-{\bm{z}}_{l1})\oplus(\hat{{\bm{z}}}_{l}-{\bm{z}}_{l2})\oplus(\hat{{\bm{z}}}_{l}-{\bm{z}}_{l3})\oplus\mathbf{\lambda}.$
(6)

At last, the implicit function (i.e., SIS decoder) can be expressed as

$\displaystyle{\bm{v}}=f({\bm{z}}_{l},\hat{\theta},\hat{\varphi}).$ (7)

###  III-D Loss Function Design

Our SIS representation defines the coordinates that are one-to-one
corresponding to the surface of 3D shapes. Thus, we can train the models in a
self-supervised manner for each vertex of 3D shapes. First, the L1
reconstruction loss of vertices is used as

$\displaystyle L_{rec}=\left\|{\bm{V}}-\hat{{\bm{V}}}\right\|_{1},$ (8)

where ${\bm{V}}$ is the ground truth vertices and $\hat{{\bm{V}}}$ is the
vertices predicted by our SIS decoders. Then, Laplacian regularization is
introduced to help the mesh reconstruction. Laplacian term is defined as the
difference between the vertex and the mean of its one-ring neighbors,
expressed as
${\bm{V}}_{i}-\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}{\bm{V}}_{j}$
where ${\bm{V}}_{i}$ is the $i$th vertex and $\mathcal{M}_{i}$ is the indices
of its one-ring neighbors of ${\bm{X}}_{i}$. We propose a Laplacian loss that
calculates the Laplacian term difference between the ground truth vertices and
the predicted vertices, expressed as

$\displaystyle
L_{lap}=\sum_{i\in\mathcal{M}}\left\|\left({\bm{V}}_{i}-\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}{\bm{V}}_{j}\right)-\left(\hat{{\bm{V}}}_{i}-\frac{1}{|\mathcal{N}_{i}|}\sum_{j\in\mathcal{N}_{i}}\hat{{\bm{V}}}_{j}\right)\right\|_{1},$
(9)

where $\mathcal{M}$ is the vertex indices of the mesh. The overall loss
function is defined as

$\displaystyle L=L_{rec}+\gamma L_{lap},$ (10)

where $\gamma=0.05$ in our experiments. During the inference phase, we can
output a 3D shape simply by inputting the spherical coordinates with a global
feature or a local feature to the SIS decoders.

##  IV EXPERIMENTS AND EVALUATION

In this section, we evaluate our SIS representation on two different 3D shape
datasets in two tasks: reconstruction task and super-resolution task. For the
reconstruction task, we input meshes with fixed topology and condition the SIS
representation with a global feature. For the super-resolution task, we input
point clouds that are randomly downsampled from meshes and condition the SIS
representation with a local feature that is assembled by a feature fusion
module based on baryccentric coordinates.

#### Datasets

In line with [10], we evaluate our model on two datasets: COMA [27] and DFAUST
[4]. COMA is a human facial dataset that consists of 12 classes of extreme
expressions from 12 different subjects. The dataset contains 20,466 3D meshes
that were registered to a common reference template with 5,023 vertices.
DFAUST is a human body dataset that collects over 40,000 real meshes,
capturing 129 dynamic performances from 10 subjects. The meshes were also
registered to a common reference topology that has 6,890 vertices. Both two
datasets are split into training and test set with a ratio of 9:1 and randomly
select 100 samples from the training set for validation. The test samples are
obtained by picking consecutive frames of length 10 uniformly at random across
the sequences. All of the 3D meshes are standardized to have a mean of 0 and
standard deviation of 1 to speed up the training.

#### Training

We use Adam [18] optimizer with learning rate 0.001 and reduce the learning
rate with decay rate 0.98 in every epoch. The batch size is 64 and total epoch
number is 200. Weight decay regularization is used for the network parameters.
We implemented the models in PyTorch and trained on the same machine with an
AMD 3700X @3.6GHz CPU and an NVIDIA RTX2080Ti GPU.

#### Architecture

As shown in Fig. 2b, we adopt the mesh encoder from [10]. The encoder has four
LSA-Conv layers with downsampling. The conv layers have channel sizes of [3,
16, 32, 64, 128] and meshes are downsampled with ratios of [4, 4, 4, 4]. A
fully connected layer outputs the latent vector of 64 dimension that
represents the 3D mesh.

For COMA dataset, as shown in Fig. 1, the template facial mesh is split into
three genus-0 meshes: left eye, right eye, and the rest part. Thus, we need
three SIS networks to represent the facial meshes. For DFAUST dataset, the
template body mesh is split into six genus-0 meshes: head, torso, left arm,
right arm, left leg, and right leg. Thus, we need six SIS networks to
represent the body meshes. Each SIS network is an MLP with a skip connection
in the middle layer. As shown in Fig. 2, the SIS encoders are conditioned with
vertices in addition to the spherical coordinates and output the corresponding
deep features. The SIS decoders are conditioned with local features in
addition to the spherical coordinates and output the vertices of 3D shapes.

###  IV-A Task 1: Reconstruction

TABLE I: Comparison of reconstruction errors for the models of LSA-small [10],
FeaStNet [30], and template deformation (TDeform) [15] when latent size
$d=64$. For a fair comparison, we adjust the channel sizes to have around the
same parameter size. ✓ represents the decoder can infer 3D shapes in an
arbitrary resolution. ✗ represents the decoder can only infer 3D shapes in a
fixed resolution of the template. The ‘time (s)’ denotes the duration to infer
the test sets.

|  |  | L2(mm)$\downarrow$ | time (s)$\downarrow$ | parm #
---|---|---|---|---|---
DFAUST | LSA-small [10] | ✗ | 3.679 | 3.992 | 547K
FeaStNet [30] | ✗ | 3.769 | 5.146 | 548K
TDeform [15] | ✓ | 6.897 | 4.391 | 549K
SIS (ours) | ✓ | 4.737 | 3.273 | 547K
COMA | LSA-small [10] | ✗ | 0.172 | 5.615 | 378K
FeaStNet [30] | ✗ | 0.208 | 9.969 | 378K
TDeform [15] | ✓ | 0.946 | 5.434 | 378K
SIS (ours) | ✓ | 0.179 | 5.357 | 378K

For the reconstruction task as shown in Fig. 2b, we compare three existing
methods: LSA-Conv, FeaStNet, and template deformation (TDeform) when the
latent space is 64. TDeform proposed by [15] uses the template as the
canonical coordinate of meshes. Similar to SIS representation, the TDeform
decoder is also built by an MLP network that predicts the deformation of the
vertices of a mesh relative to the template vertices. During the inference
phase, we can provide a higher-resolution template to predict 3D shapes that
have the same resolution as the template. Table I shows the quantitative
results. For a fair comparison, we adjust the channel sizes for each methods
to have around the same model size. For methods that can infer 3D shapes in an
arbitrary resolution (labeled as ✓in Table I), our SIS representation
outperforms TDeform in both DFAUST and COMA datasets. For COMA dataset, our
representation even performs better than FeaStNet that only works in a fixed
resolution. In terms of time comlexity, the proposed SIS is the most time-
efficient compared with other methods since SIS networks are simply MLPs.

Note that, we split both the facial template and body template into multiple
genus-0 submeshes and each submesh requires an SIS network. In order to
control the overall model size to be around the same with other methods, the
parameter size for each SIS network is small. As shown in Table I, we split
more parts for the body template than for the facial template, thus, each body
part has a smaller SIS network and only has 5 or 6 layers with 131 channel
size, resulting in larger errors in DFAUST dataset than COMA dataset compared
to other methods. However, even we need an extra SIS network for the eyes in
COMA dataset, our SIS representation is marginally on par (0.179 vs. 0.172)
with LSA-small that is the current best convolutional operation designed for
meshes. Even though controlling the overall model size to be the same with
other methods is not favorable for our setting, our SIS representation
consistently outperforms TDeform that uses one but deeper and larger MLP
network. For TDeform, the input could be any $xyz$ point in the volumetric
space while only the points of template vertices are trained with labels.
Thus, most of samples (except for the template vertices) are not trained for
the implicit function of TDeform, i.e., undersampling occurs.

###  IV-B Task 2: Super-resolution

Figure 3: Comparison of reconstruction errors between our SIS representation
and BCI (barycentric interpolation) for the super-resolution task. We train
the models with 1,000 input points and infer the models with input points of
500, 1,000, and 1,500.

‘

Figure 4: Qualitative results of the super-resolution task. The per-vertex
Euclidean errors produced by our SIS representation and BCI are visualized in
colormap. The input point cloud has 1,000 points that are randomly sampled
from the ground truth mesh. The left and right are some examples wihh varisou
facial expressions and body poses from the test sets of COMA and DFAUST
datasets.

For the super-resolution task as shown in Fig. 2c, we randomly sample 1,000
points from a mesh as the input to train our models in a self-supervised
manner in DFAUST and COMA datasets. We compare our method with a traditional
algorithm: barycentric interpolation (BCI). BCI interpolates the vertex of a
given spherical coordinate based on the barycentric coordinates that are
calculated from the triangular face on the sphere. For instance, when the
triangular vertices are $[{\bm{v}}_{1},{\bm{v}}_{2},{\bm{v}}_{3}]$ and the
barycentric coordinates are $[\lambda_{1},\lambda_{2},\lambda_{3}]$ where
$\sum_{i=1}^{3}\lambda_{i}=1$, the interpolated vertex is expressed as
${\bm{v}}=\lambda_{1}{\bm{v}}_{1}+\lambda_{2}{\bm{v}}_{2}+\lambda_{3}{\bm{v}}_{3}$.

We evaluate our approach and BCI with three different numbers of input points:
500, 1,000, and 1,500. As shown in Fig. 3, our SIS representation consistently
outperforms BCI in both DFAUST and COMA datasets for all the different numbers
of input points, which demonstrates the robustness of our SIS representation.
The qualitative results presented in Fig. 4 also show that our approach
produces smaller errors than BCI for both DFAUST and COMA datasets in various
body poses and facial expressions.

###  IV-C Ablation Study

For the super-resolution task, we design a feature fusion module to ensemble
the deep features for the local feature of a given spherical coordinate. To
evaluate the effectiveness of the feature fusion module, we conduct an
ablation study where we simply use the coarse deep feature
$\hat{{\bm{z}}}_{l}$ (Eq. 5) as the local feature of a given spherical
coordinate without the feature fusion module, denoted as _SIS_w/o_ in Figure
5. For both the COMA and DFAUST datasets, our SIS representation with the
feature fusion module outperforms _SIS_w/o_ consistently with different input
points. This is berceuse the feature fusion module considers the edges between
the coarse deep feature with the deep features of the triangular vertices and
provides more local structure around the spherical coordinate. Thus, our SIS
representation with the feature fusion module can generate 3D shapes with more
details.

Figure 5: Ablation study for of the feature fusion module in the super-
resolution task. “SIS_w/o” means we use the coarse feature in Eq. 6 as the
input of the SIS decoder without the feature fusion module. We train the
models with 1,000 input points and infer the models with input points of 500,
1,000, and 1,500.

##  V CONCLUSION AND DISCUSSIONS

###  V-A Conclusion

We propose to learn the continuous representation for meshes, which is
fulfilled by our devised spherical implicit surface (SIS) technique. SIS
builds a bridge between the discrete and continuous representation in mesh and
can naturally exploit the information provided in different resolutions. To
share knowledge across samples, we condition the SIS representation with a
global feature or a set of local features of a mesh. We show that this
continuous representation technique can be effectively applied for downstream
tasks like reconstruction and super-resolution of 3D shapes.

###  V-B Limitations

The SIS representation for meshes is similar to the implicit function for
images [28]. When the resolution of a mesh is too low, the SIS representation
may overfit to the small amount of training samples and cannot generalize well
to the whole surface of the mesh. Thus, high-resolution meshes are more
favorable to train an SIS network. Furthermore, the experimented datasets may
not fully reflect the challenges in real-world scenarios.

###  V-C Future works

In this work, we split a mesh template into multiple genus-0 submeshes and
train an independent SIS network for each submesh. In the future, we can
create a shared SIS network for all the submeshes to reduce the model size.
Furthermore, currently, we simply encode the spherical coordinates with
Fourier feature mapping. More advance coordinate encoding methods [22] can be
integrated to our SIS representation.

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