Probabilistic type 2 Bernoulli and Euler polynomials

Li Chen; Dmitry V. Dolgy; Taekyun Kim; Dae San Kim; Li Chen; Dmitry V. Dolgy; Taekyun Kim; Dae San Kim

doi:10.3934/math.2024696

AIMS Mathematics

2024, Volume 9, Issue 6: 14312-14324. doi: 10.3934/math.2024696

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Probabilistic type 2 Bernoulli and Euler polynomials

1.
School of Mathematics, Xi'an University of Finance and Economics, Xi'an, China
2.
Kwangwoon Global Education Center, Kwangwoon University, Seoul 139-701, Republic of Korea
3.
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
4.
School of Science, Xi'an Technological University, Xi'an 710021, Shaanxi, China
5.
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

Received: 05 March 2024 Revised: 26 March 2024 Accepted: 09 April 2024 Published: 19 April 2024
MSC : 11B68, 11B73, 11B83

Assume that the moment-generating function of the random variable $ Y $ exists in a neighborhood of the origin. The aim of this paper is to investigate the probabilistic type 2 Bernoulli polynomials associated with $ Y $ and the probabilistic type 2 Euler polynomials associated with $ Y $, along with the probabilistic type 2 cosine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 sine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 cosine-Euler polynomials associated with $ Y $, and the probabilistic type 2 sine-Euler polynomials associated with $ Y $. We deal with their properties, related identities and explicit expressions.
Citation: Li Chen, Dmitry V. Dolgy, Taekyun Kim, Dae San Kim. Probabilistic type 2 Bernoulli and Euler polynomials[J]. AIMS Mathematics, 2024, 9(6): 14312-14324. doi: 10.3934/math.2024696

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Abstract

Assume that the moment-generating function of the random variable $ Y $ exists in a neighborhood of the origin. The aim of this paper is to investigate the probabilistic type 2 Bernoulli polynomials associated with $ Y $ and the probabilistic type 2 Euler polynomials associated with $ Y $, along with the probabilistic type 2 cosine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 sine-Bernoulli polynomials associated with $ Y $, the probabilistic type 2 cosine-Euler polynomials associated with $ Y $, and the probabilistic type 2 sine-Euler polynomials associated with $ Y $. We deal with their properties, related identities and explicit expressions.

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