On a third kind of characteristic numbers of the spheroidal functions

A set of real positive numbers γ_αn(c) was found recently to be associated with the spheroidal functions of real order α > -1 as a consequence of their double orthogonality on (-1, 1) and (- ∞, ∞). In the range -1 < α < 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 < α < 0 but it leads to computational formulas for γ_αn(c) that appear from numerical results to be valid for all real α > -1.