A modification of the Prudnikov and Laguerre polynomials

TL;DR

This paper introduces a new two-parameter family of orthogonal polynomials generalizing Prudnikov and Laguerre polynomials, analyzing their properties and special cases with potential applications in mathematical analysis.

Contribution

The paper defines and investigates a novel two-parameter sequence of orthogonal polynomials related to modified Bessel functions, extending classical polynomial families with new parameter-dependent properties.

Findings

Orthogonal polynomials with respect to a specific weight function are constructed.

Special cases recover Prudnikov and Laguerre polynomials.

A one-parameter subset is analyzed in detail.

Abstract

A two-parameter sequence of orthogonal polynomials $\{P_n( x; \lambda, t)\}_{n\ge 0}$ with respect to the weight function $x^\alpha e^{- \lambda x} \rho_\nu(x t),\ \alpha > -1,\ \lambda, t \ge 0, \ \rho_{\nu}(x)= 2 x^{\nu/2} K_\nu(2\sqrt x),\ x >0, \nu \ge 0$, where $K_\nu(z)$ is the modified Bessel function, is investigated. The case $\lambda=0$ corresponds to the Prudnikov polynomials and $t=0$ is related to the Laguerre polynomials. A special one-parameter case $\{P_n( x; 1-t, t)\}_{n\ge 0},\ t \in [0,1]$ is analyzed as well.