Integral Formulas for the Noncentral Tanny-Dowling Polynomials

TL;DR

This work derives integral identities for the noncentral Tanny-Dowling polynomials $\widetilde{F}_{m,a}(n;x)$ and connects them to classical geometric polynomials and Bernoulli polynomials. It uses exponential generating functions and integral transforms to establish Kellner-type integral formulas, Worpitzky-type expansions in terms of noncentral Whitney numbers $\widetilde{W}_{m,a}(n,k)$, and Boyadzhiev-type relations for the noncentral polynomials. A key result is the integral formula $\int_{-1}^{0} \widetilde{F}_{m,a}(n; m x)\,dx = m^{n} B_{n}\left(-\frac{a}{m}\right)$, along with a Bernoulli-polynomial expansion $B_n\left(-\frac{a}{m}\right)=\sum_{k=0}^{n} k! \widetilde{W}_{m,a}(n,k)\frac{(-1)^k}{m^{n-k}(k+1)}$, and the representation $\widetilde{F}_{m,a}(n;x)=\int_{0}^{\infty} \widetilde{D}_{m,a}(n;x\lambda) e^{-\,}\lambda\,d\lambda$. The results specialize to known formulas for $m=1,a=0$, linking to Boyadzhiev’s exponential generating function and Kellner’s identity, and extend the scope to Dowling polynomials and noncentral variants with potential combinatorial interpretations.

Abstract

In this paper, we established some integral formulas for and involving the noncentral Tanny-Dowling polynomials. These formulas are shown to be generalizations of some known results on the classical geometric polynomials.

Theorems & Definitions (11)

• Theorem 1
• proof
• Remark 2
• Corollary 3
• Remark 4
• Theorem 5
• proof
• Remark 6
• Theorem 7
• proof