Entropic integrals of orthogonal hypergeometric polynomials with general supports

Abstract

The Boltzmann-Shannon information entropy of probability measures which involve the continuous hypergeometric-type polynomials {pn(x)}, orthogonal with respect to a general weight function ω(x), is determined by two integral quantities: one with kernel pn2(x)ω(x) ln pn2(x), called as entropy of the polynomial pn(x), and another one with kernel pn2(x)ω(x) ln ω(x). Here, an explicit expression for the latter quantity, and for a broader family of related integrals, is obtained in terms only of the second-order differential equation satisfied by the involved polynomials. For illustration, the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type (Hermite, Laguerre, and Jacobi).