A new construction of probabilistic Hermite polynomials with their certain applications

Burcu Doğruer Doğan; Mehmet Acikgoz; Ayse Karagenc; Serkan Araci; Burcu Doğruer Doğan; Mehmet Acikgoz; Ayse Karagenc; Serkan Araci

doi:10.3934/nhm.2026041

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 997-1016. doi: 10.3934/nhm.2026041

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Research article

A new construction of probabilistic Hermite polynomials with their certain applications

1.
Department of Mathematics, Faculty of Science and Arts, Gaziantep University, TR-27310 Gaziantep, Türkiye
2.
Department of Information Security Technology, Cappadocia Vocational School, Cappadocia University, TR-50420 Nevşehir, Türkiye
3.
Department of Computer Engineering, Faculty of Engineering, Hasan Kalyoncu University, TR-27010 Gaziantep, Türkiye

Received: 05 February 2026 Revised: 17 April 2026 Accepted: 24 April 2026 Published: 14 May 2026

Our aim of this paper was to introduce a new construction of probabilistic Hermite polynomials based on moment generating functions. By using this generating function, we derived several new relations and formulas among the aforementioned polynomials and other types of probabilistic special number sequences and polynomials, such as the probabilistic Stirling numbers of the second kind, probabilistic Bernoulli polynomials of higher order, probabilistic Bernstein polynomials, and probabilistic Euler polynomials of higher order. By selecting special random variables, including Poisson, Uniform, Gamma, Geometric, Exponential, and Normal random variables, we showed that the generating function of probabilistic Hermite polynomials yields distinct and unique generating functions, which lead to new relations among other types of special numbers and polynomials, as presented in the application section of this paper.
Citation: Burcu Doğruer Doğan, Mehmet Acikgoz, Ayse Karagenc, Serkan Araci. A new construction of probabilistic Hermite polynomials with their certain applications[J]. Networks and Heterogeneous Media, 2026, 21(3): 997-1016. doi: 10.3934/nhm.2026041

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Abstract

Our aim of this paper was to introduce a new construction of probabilistic Hermite polynomials based on moment generating functions. By using this generating function, we derived several new relations and formulas among the aforementioned polynomials and other types of probabilistic special number sequences and polynomials, such as the probabilistic Stirling numbers of the second kind, probabilistic Bernoulli polynomials of higher order, probabilistic Bernstein polynomials, and probabilistic Euler polynomials of higher order. By selecting special random variables, including Poisson, Uniform, Gamma, Geometric, Exponential, and Normal random variables, we showed that the generating function of probabilistic Hermite polynomials yields distinct and unique generating functions, which lead to new relations among other types of special numbers and polynomials, as presented in the application section of this paper.

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