Information entropy of Gegenbauer polynomials and Gaussian quadrature

Abstract

In a recent paper (Buyarov V S, Lopez-Artes P, Martinez-Finkelshtein A and Van Assche W 2000 J. Phys. A: Math. Gen. 33 6549–60), an efficient method was provided for evaluating in closed form the information entropy of the Gegenbauer polynomials C(λ)n(x) in the case when λ = l . For given values of n and l, this method requires the computation by means of recurrence relations of two auxiliary polynomials, P(x) and H(x), of degrees 2l − 2 and 2l − 4, respectively. Here it is shown that P(x) is related to the coefficients of the Gaussian quadrature formula for the Gegenbauer weights wl(x) = (1 − x2)l−1/2, and this fact is used to obtain the explicit expression of P(x). From this result, an explicit formula is also given for the polynomial S(x) = limn→∞ P(1 − x/(2n2)), which is relevant to the study of the asymptotic (n → ∞ with l fixed) behaviour of the entropy.