Recurrence relations for the coefficients of asymptotic expansions for prolate spheroidal and ellipsoidal wave equations (Q791725)

The prolate spheroidal wave equation is transformed into a form containing a parameter c, such that for \(c=0\) it is satisfied by a Jacobi polynomial. For \(c\simeq 0\), the equation is solvable by a perturbation series, proceeding by powers of c; this paper derives a recurrence relation between the coefficients of powers of c in this series. The same procedure is then applied to the prolate spheroidal wave equation in a different form, such that for \(c=\infty\) it is satisfied by an associated Laguerre polynomial; the perturbation series not proceeds in powers of \(c^{-1}\). Finally, the ellipsoidal wave equation is treated in the same way, the perturbation series being that of \textit{H. J. W. Müller} [Math. Nachr. 31, 89-101 (1966; Zbl 0147.059)]. In this part of the paper misprints are numerous enough to make reference to Müller's paper necessary.