On a Third Kind of Characteristic Numbers of the Spheroidal Functions

A set of real positive numbers gamma(alphan)(c) was found recently to be associated with the spheroidal functions of real order alpha > -1 as a consequence of their double orthogonality on (-1, 1) and (- infinity, infinity). In the range -1 < alpha < 0 these numbers are shown to be determined by the eigenvalues of a new integral equation for the spheroidal functions. Thus they represent a third kind of characteristic numbers. The new equation ceases to be an integral equation above the range -1 < alpha < 0 but it leads to computational formulas for gamma(alphan)(c) that appear from numerical results to be valid for all real alpha > -1.