A particular case of Truesdell's unified theory of special functions

Solution 1:

The expression with the puzzling $\rm\:c(v)\:$ is Norlund's principal solution of the difference equation $\rm \mathop\Delta\limits_{\alpha}\ \mathop {\mathrm S}\limits_{{\alpha _0}}^\alpha h(v) dv = h(a).\: $ As Truesdell mentions in Appendix II, one can find an exposition of this in Chapter 8 of the classic The Calculus of Finite Differences by Milne-Thomson.

As I have mentioned previously here, Willard Miller showed that Truesdell's method is essentially Lie-theoretic. See his freely available book Lie theory and Special Functions, 1968. There he also shows that, similarly, the Schroedinger-Infeld-Hull ladder / factorization method (a powerful tool widely exploited by physicists to compute eigenvalues, recurrence relations, etc. for solutions of second order ODEs) is essentially equivalent to the representation theory of four local Lie groups. Nowadays it is a special case of Lie-theoretic symmetry methods used for separation of variables in partial differential equations (a major theme in the group-theoretic approach towards a unified theory of special functions).