Nicholson-Type Integrals for Products of Gegenbauer Functions and Related Topics

Abstract

Nicholson's integral gives a generalization of the relation sin2x + cos2x = 1 to the case of Bessel functions. This integral expresses the sum of the squares of the Bessel functions of the first and second kind, [J2v(x)+Y2v(x)] as an integral over a hyperbolic Bessel function, with the integrand positive. We present an analogous result for general Gegenbauer and Legendre functions derived from a new representation for the product of two Gegenbauer functions of the second kind, Dλ(α) (x). The result expresses the sum of the squares of the Gegenbauer functions on the cut, [(C(α)λ(x))2 + (D(α)λ(x))2], as an integral over a Gegenbauer function of the second kind, with the integrand again positive. The result for the ordinary Legendre functions is quite simple