[Paper Review] Some identities of symmetry for q-Bernoulli polynomials under symmetric group of degree n

Dae San Kim, Taekyun Kim|arXiv (Cornell University)|Apr 15, 2015

Advanced Mathematical Identities1 references8 citations

TL;DR

This paper establishes new identities of symmetry for Carlitz's $q$-Bernoulli polynomials under the action of the symmetric group $S_n$ by leveraging $p$-adic $q$-integrals on $\mathbb{Z}_p$. The key contribution is a symmetric expression in $w_1, \dots, w_n$ that remains invariant under any permutation $\sigma \in S_n$, derived via generating functions and $q$-integral identities involving $q$-analogues of binomial expansions and $p$-adic measures.

ABSTRACT

In this paper, we give some new identities of symmetry for q-Bernoulli polynomials under the symmetric group of degree n arising from p-adic q-integrals on Zp.

Motivation & Objective

• To derive symmetric identities for $q$-Bernoulli polynomials under the full symmetric group $S_n$ of degree $n$.
• To extend known symmetry identities for $q$-Bernoulli polynomials beyond the $n=2$ case to arbitrary $n \geq 2$.
• To establish invariance of certain $q$-integral expressions under permutations of the parameters $w_1, \dots, w_n$.
• To unify and generalize previous results on $q$-Bernoulli polynomials using $p$-adic $q$-integration and symmetric function theory.

Proposed method

• Utilizes the $p$-adic $q$-integral on $\mathbb{Z}_p$ as defined by Kim, expressing $q$-Bernoulli polynomials as $\beta_{n,q}(x) = \int_{\mathbb{Z}_p} [x+y]_q^n d\mu_q(y)$.
• Applies a multivariate generating function approach involving exponential generating functions with $q$-exponential terms and $p$-adic measures.
• Employs a change of variables and summation over residue classes modulo $w_n$ to decompose the $p$-adic integral into a sum over $w_n$-blocks.
• Derives a key identity (2.2) showing invariance under permutation $\sigma \in S_n$ by reindexing and reordering the parameters $w_1, \dots, w_n$.
• Uses the $q$-binomial expansion and $q$-analogue identities to express $[y + w_n x + w_n \sum \frac{k_j}{w_j}]_{q^{w_1 \cdots w_{n-1}}}^m$ in terms of $[y + w_n x]_q^l$ and $q$-powers.
• Introduces the auxiliary function $T_{m,q}(w_1,\dots,w_{n-1} \mid l)$ to encapsulate symmetric sums over $k_j$-indices, enabling permutation invariance.

Experimental results

Research questions

• RQ1 How can symmetric identities for $q$-Bernoulli polynomials be generalized from $n=2$ to arbitrary $n \geq 2$ under the symmetric group $S_n$?
• RQ2 What conditions ensure invariance of $q$-integral expressions involving $w_1, \dots, w_n$ under permutation of the parameters?
• RQ3 Can the $p$-adic $q$-integral representation of $q$-Bernoulli polynomials be used to derive symmetric identities via multivariate generating functions?
• RQ4 What role do $q$-analogues of binomial coefficients and $q$-exponential functions play in constructing symmetric identities?
• RQ5 How does the structure of the $q$-integral transform under change of variables involving multiple scaling parameters $w_1, \dots, w_n$?

Key findings

• The expression $\left[\prod_{j=1}^{n-1}w_{\sigma(j)}\right]_{q}^{m-1}\prod_{l=1}^{n-1}\sum_{k_l=0}^{w_{\sigma(l)}-1} q^{w_{\sigma(n)}\sum_{j=1}^{n-1}\left(\prod_{i \neq j} w_{\sigma(i)}\right)k_j} \times \beta_{m,q^{w_{\sigma(1)}\cdots w_{\sigma(n-1)}}}\left(w_{\sigma(n)}x + w_{\sigma(n)}\sum_{j=1}^{n-1}\frac{k_j}{w_{\sigma(j)}}\right)$ is invariant under any $\sigma \in S_n$.
• The symmetric identity is established via the invariance of the $p$-adic $q$-integral expression (2.2) under permutation of the parameters $w_1, \dots, w_n$.
• The derived symmetric identity generalizes known results for $n=2$ to arbitrary $n$, showing that the structure of $q$-Bernoulli polynomials respects full symmetric group action.
• The function $T_{m,q^{w_n}}(w_1,\dots,w_{n-1} \mid l)$ is symmetric in $w_1, \dots, w_{n-1}$, ensuring the entire expression remains invariant under $S_n$.
• The key identity (2.7) decomposes the $q$-integral into a sum involving $\beta_{l,q^{w_1\cdots w_{n-1}}}(w_n x)$, with coefficients depending on $q$-binomial and $q$-power terms.
• The final symmetric identity (Theorem 2.3) confirms that the entire expression remains unchanged under any permutation $\sigma \in S_n$, proving the symmetry of the $q$-Bernoulli polynomial identities.