Representations by probabilistic Bernoulli and degenerate Bernoulli polynomials

2025/10/24 by Kim, Dae san, Kim, Taekyun
#05A19 #05A40 #11B68 #11B73 #11B83 #60-08 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.21558

Abstract

We investigate the representation of arbitrary polynomials using probabilistic Bernoulli and degenerate Bernoulli polynomials associated with a random variable Y, whose moment generating function exists in a neighborhood of the origin. In addition, this paper explores the problem of representing arbitrary polynomials in terms of their higher-order counterparts. We develop explicit formulas for those representations with the help of umbral calculus and illustrate our results for several discrete and continuous random variables Y.

Citations

• Probabilistic degenerate Stirling numbers of the first kind and their applications
• Representing polynomials by degenerate Bernoulli polynomials
• Probabilistic Stirling numbers of the second kind and applications
• A probabilistic generalization of the Stirling numbers of the second kind
• Hodge integrals and Gromov-Witten theory

Related

• Representations by probabilistic Frobenius-Euler and degenerate Frobenius-Euler polynomials
• Probabilistic degenerate Bernstein polynomials
• Probabilistic Representation of Bernoulli, Euler and Carlitz Hermite Polynomials
• Probabilistic degenerate Fubini polynomials associated with random variables
• Representing polynomials by degenerate Bernoulli polynomials