Recurrence relations for the indefinite integrals of the associated Legendre functions

The indices n and m are treated as nonnegative integers throughout, and a is a constant usually taken as zero or minus one. The final expressions are given below by (12) and (17) and in a particular normalized form in (12) and (17). It is unlikely these relations are new, however we have not been able to find them in the literature, and their absence is conspicuous in such classical texts as [1], [4].* The need for such relations occurs in geodesy studies, [3], [5]. We first list for easy reference some well-known relations, [1, p. 1325]: