On New Probabilistic Hermite Polynomials

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On New Probabilistic Hermite Polynomials

Temitope O. Alakija1; Ismaila S. Amusa2; Bolanle O. Olusan3; Ademola A. Fadiji 4
Department of Statistics1&4, Department of Mathematics2&3, Yaba College of Technology, Lagos, Nigeria
DOI: https://doi.org/10.51584/IJRIAS.2023.8702
Received: 01 June 2023; Revised: 23 June 2023; Accepted: 29 June 2023; Published: 26 July 2023


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Abstract: – In the theory of differential equation and probability, Probabilistic Hermite polynomials H<sub>r</sub>(x) = {r=0,1,2,…,n} are the polynomials obtained from derivatives of the standard normal probability density function (pdf) of the form α(x)=1/√2π e^(-1/2 x^2 ). These polynomials played an important role in the Gram-Charlier series expansion of type A and the Edgeworth’s form of the type A series (see [18]).
In this paper, we obtained new Probabilistic Hermite polynomials by considering a standard normal distribution with probability density function (pdf) given as β(x)=1/(2√π) e^(-1/4 x^2 ). The generating function, recurrence relations and orthogonality properties are studied. Finally, a differential equation governing these polynomials was presented which enables us to obtain the expression of the polynomial in a closed form.

Keywords: Generating Function, recurrence relation, differential equation, Power series, Orthogonality.

I. Introduction and Preliminary

Special functions and polynomials are solutions of special differential equations; they appear in mathematics, statistics, Lie group theory, and number theory. Probabilistic Hermite polynomials H<sub>r</sub>(x) are also special polynomials that occur in the theory of advanced statistics and are given by the series expansion