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
$$K_{n_1,n_2}^{({\alpha }_1,{\alpha }_2)}(x,k,p)=(-1{)}^{n_1+n_2}{e}^{p{x}^k}[\prod _{j=1}^2{x}^{-\alpha }\frac{d^{n}j}{d{x}^{n_j}}(x{)}^{{\alpha }_j+n_j}]e^{-p{x}^k},$$
and
$$M_{n_1,n_2}^{({\alpha }_0,p_1,p_2)}(x,k)=\frac{(-1{)}^{n_1+n_2}}{p_1^{n_1}{p}_2^{p_2}}{x}^{-{\alpha }_0}[\prod _{j=1}^2{e}^{p_j{x}^k}\frac{d^{n}j}{d{x}^{n_j}}d{x}^{n_j}{e}^{-p_j{x}^k}]x^{n_1+n_2+{\alpha }_0},$$
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.