Degenerate Sheffer sequences and λ-Sheffer sequences

By Dae San Kim, Taekyun Kim. 2021

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Abstract In the 1970s, Gian-Carlo Rota laid a rigorous foundation for umbral calculus which is based on the modern concepts like a linear functional, a differential operator and an adjoint. The motivation for this paper starts from the question that what if the usual exponential function in the generating function of a Sheffer sequence is replaced by degenerate exponential functions. The sequences thus defined are called λ-Sheffer sequences. It may be said that this question is very natural in view of the regained recent interests in degenerate special numbers and polynomials. It turns out that these correspond to replacing the linear functional and the differential operator respectively by λ-linear functionals and λ-differential operators. In this article, we give an account of λ-Sheffer sequences and degenerate Sheffer sequences, and illustrate them with many examples.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e0e53588e90cccef2f9b2074c1956ef88b795a11