Generalized Hermite polynomials

Gospava B. Djordjevi\'c

Abstract: We consider a new generalization of the classical Hermite polynomials and prove the basic characteristics of such polynomials $h^{\lambda}_{n,m}(x)$ (the generating function, an explicit representation, some recurrence relations, and the corresponding differential equation). For $m=2$, the polynomial $h^{\lambda}_{n,m}(x)$ reduces to $H_n(x,{\lambda})/n!$, where $H_n(x,{\lambda})$ is the Hermite polynomial with a parameter. For $\lambda = 1$, $h^l_{n,2}(x) = H_n(x)/n!$, where $H_n(x)$ is the classical Hermite polynomial. Taking $\lambda=1$ and $n=mN+q$, where $N=[n/m]$ and $0\leq q\leq m-1$, we introduce the polynomials $P_N^{(m,q)}(t)$ by $h^l_{n,m}(x) = (2x)^q P_N^{(m,q)}((2x)^m)$, and prove that they satisfy an $(m+1)$-term linear recurrence relation.

Classification (MSC2000): 33C55

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