Entropic integrals of orthogonal hypergeometric polynomials with general supports (Q1571026)

Two integrals arise in the calculation of the Boltzman-Shannon entropy of probability measures associated with continuous hypergeometric polynomials \(\{p_n(x)\}\) orthogonal with respect to a general weight function \(\omega(x)\) on an interval \(J\). The authors focus on the calculation of the integral \(I_n=-\int_J p^2_np^2_n(x) \omega(x) \ln\omega (x)dx\) over intervals with compact and noncompact support. They derive a recursion relation and closed analytic expressions for a family of integrals that includes \(I_n\). Their formulae only involve the coefficients of the differential equation satisfied by the polynomials \(p_n(x)\). The formulae are applied to evaluate \(I_n\) for the Hermite, Laguerre and Jacobi polynomials.