On New Probabilistic Hermite Polynomials

In the theory of differential equation and probability, Probabilistic Hermite polynomials Hr(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.