Some Generalizations of Multiple Laguerre Polynomials via Rodrigues Formula

Abstract

This paper aims to provide a systematic investigation of the family of polynomials generated by the Rodrigues’ formulas
\[K_{n_1,n_2}^{(\alpha_1, \alpha_2)}(x, k,p) = (-1)^{n_1+n_2} e^{px^k}[\prod\limits_{j=1}^2x^{-\alpha}\frac{d^nj}{dx^{n_j}} (x)^{\alpha_j+n_j}]e^{-px^k},\]
and
\[M_{n_1,n_2}^{(\alpha_0,p_1,p_2)}(x, k) = \frac{(-1)^{n_1+n_2}}{p_1^{n_1}p_2^{p_2}}x^{-\alpha_0}[\prod\limits_ {j=1}^{2}e^{p_jx^k}\frac{d^nj}{dx^{n_j}}{dx^{n_j}}e^{-p_jx^k}]x^{n_1+n_2+\alpha_0},\]
These polynomials include the multiple Laguerre and the multiple Laguerre-Hahn polynomials, respectively. The explicit forms, certain operational formulas involving these polynomials with some applications, and linear generating functions for \(K_{n_1,n_2}^{(\alpha_1, \alpha_2)}(x, k,p)\) and \(M_{n_1,n_2}^{(\alpha_0,p_1,p_2)}(x, k)\) are obtained.