A modification of the Prudnikov and Laguerre polynomials

Abstract

A two-parameter sequence of orthogonal polynomials $\{P_n(x;\lambda ,t){\}}_{n\ge 0}$ with respect to the weight function $x^{\alpha }{e}^{-\lambda x}{\rho }_{\nu }(xt),\alpha >-1,\lambda ,t\ge 0,{\rho }_{\nu }(x)=2x^{\nu /2}{K}_{\nu }(2\sqrt{x}),x>0,\nu \ge 0$, 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.