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λ-q-Sheffer sequence and its applications

Kim Taekyun, Kim Dae San, Kim Hye Kyung

Demonstratio Mathematica

2022

Vol. 55, nr 1

843--865

Recently, Kim-Kim [J. Math. Anal. Appl. 493 (2021), no. 1] introduced the degenerate Sheffer sequence and λ-Sheffer sequence. The purpose of this article is to study λ-q-Sheffer sequence and the degenerate q-Sheffer sequence, which are derived from the view point of degenerate umbral calculus and investigate some properties related to those sequences. In addition, we give some new identities associated with q-special polynomials arising from our investigation.

Fully degenerate Bernoulli numbers and polynomials

Kim Taekyun, Kim Dae San, Park Jin-Woo

Demonstratio Mathematica

2022

Vol. 55, nr 1

604--614

The aim of this article is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Zp . We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r -Stirling numbers of the second kind, and the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials.

New results on the q-generalized Bernoulli polynomials of level m

Urieles Alejandro, Ortega María José, Ramírez William, Vega Samuel

Demonstratio Mathematica

2019

Vol. 52, nr 1

511--522

This paper aims to show new algebraic properties from the q-generalized Bernoulli polynomials [wzór] of level m, as well as some others identities which connect this polynomial class with the q-generalized Bernoulli polynomials of level m, as well as the q-gamma function, and the q-Stirling numbersof the second kind and the q-Bernstein polynomials.