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

D. V. Dolgy, T. Kim+2 more|arXiv (Cornell University)|Aug 1, 2016

Advanced Mathematical Identities11 references3 citations

TL;DR

This paper derives new identities of symmetry for degenerate $q$-Bernoulli polynomials using the $p$-adic $q$-integral on $\mathbb{Z}_p$. By leveraging generating functions and $p$-adic integration, the authors establish symmetric identities under permutations of parameters $w_1, \dots, w_n$, generalizing earlier results on $q$-Bernoulli polynomials and extending them to the degenerate $q$-setting via $\lambda$-deformation and $q$-integers.

ABSTRACT

Recently, Kim-Kim Introduced some interesting identities of symmetry for q-Bernoulli polynomials under symmetry group of degree n. In this paper, we study the degenerate q-Euler polynomials and derive some identities of symmetry for these polynomials arising from the p-adic q-integral on Zp.

Motivation & Objective

• To extend known symmetry identities for $q$-Bernoulli polynomials to the degenerate $q$-Bernoulli case.
• To investigate symmetric properties of degenerate $q$-Bernoulli polynomials under the action of the symmetric group of degree $n$.
• To derive explicit symmetric identities using the $p$-adic $q$-integral representation of the polynomials.
• To generalize previous results on $q$-Bernoulli polynomials by incorporating degeneracy parameters $\lambda$ and $q$-integers.

Proposed method

• Utilizes the $p$-adic $q$-integral on $\mathbb{Z}_p$ to define degenerate $q$-Bernoulli polynomials via generating functions.
• Applies the $\lambda$-deformation to the $q$-integer $[x]_q$, leading to the generating function $\int_{\mathbb{Z}_p} (1+\lambda t)^{\frac{1}{\lambda}[x+y]_q} d\mu_q(y)$.
• Employs the $p$-adic $q$-integral with scaled measures $\mu_{q^{w_1 \cdots w_{n-1}}}$ to analyze symmetric structures across $n$ parameters.
• Uses the Stirling numbers of the first kind $S_1(m,l)$ to expand powers of $q$-integers and relate them to $p$-adic integrals.
• Introduces a symmetric function $K_{n,q^{w_n}}(w_1,\dots,w_{n-1}|i,t)$ encoding weighted sums over $k_j$-indices to capture permutation invariance.
• Applies permutation invariance over $w_1, \dots, w_n$ to prove that the resulting symmetric expressions remain unchanged under any $\sigma \in S_n$.

Experimental results

Research questions

• RQ1 How do symmetric identities for $q$-Bernoulli polynomials extend to the degenerate $q$-Bernoulli case with parameter $\lambda$?
• RQ2 What symmetric identities emerge when the $p$-adic $q$-integral is applied to degenerate $q$-Bernoulli polynomials with multiple scaling parameters $w_1, \dots, w_n$?
• RQ3 Can the symmetric structure of the identities be preserved under arbitrary permutations of the parameters $w_1, \dots, w_n$?
• RQ4 What role do $q$-integers and $p$-adic measures play in constructing symmetric identities for degenerate $q$-Bernoulli polynomials?

Key findings

• The symmetric identities for degenerate $q$-Bernoulli polynomials are invariant under any permutation $\sigma$ of the parameters $w_1, \dots, w_n$, as shown in Theorem 2.3.
• The expression $\beta_{m,\frac{\lambda}{[w_1\cdots w_{n-1}]_q},q^{w_1\cdots w_{n-1}}}(w_n x + \sum_{j=1}^{n-1} \frac{w_n}{w_j}k_j)$ is symmetric in $w_1, \dots, w_n$ when averaged over $k_j$-sums.
• The symmetric function $K_{n,q^{w_n}}(w_1,\dots,w_{n-1}|p-s,s)$ captures the weighted sum over $k_j$-indices and ensures permutation invariance.
• The final symmetric identity involves a triple sum over $p,s,l$ with coefficients $S_1(m,p)$, binomial coefficients, and $q$-powers, all invariant under $\sigma \in S_n$.
• The result generalizes earlier symmetry identities for $q$-Bernoulli polynomials by incorporating degeneracy via $\lambda$ and $q$-integers.
• The identity holds for all $m \geq 0$ and positive integers $w_1, \dots, w_n$, with the symmetric structure preserved under any permutation of the $w_i$.