A new construction of probabilistic Hermite polynomials with their certain applications

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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.

Keywords

random variable / moment generating functions / probabilistic Hermite polynomials / probabilistic Stirling numbers of the second kind / probabilistic Bernoulli polynomials of higher order / probabilistic Bernstein polynomials

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