Closed analytical expressions for some useful sums and integrals involving Legendre functions (Q1097994)

Interesting formulae for (finite or infinite) sums involving products of two Legendre functions of the second kind (of half-integer order) and of integrals with one function of this type are given. The Whipple relation leads to equivalent results for associated Legendre functions of the first kind. As usually, the relations have been found by comparison of different forms of the solution to boundary value problems for partial differential equations, here in the theory of certain vector magnetic potentials, and the necessary convergence proofs have been given. The formulae are lacked in the common mathematical handbooks, but they are closely related to known representations. Thus, we may conject that they (and the known relations) are examples for more general facts to be found out henceforth. The recent work is a clearly and carefully made exposition. However, are the formulae new? - always a difficult task in this subject. The author believes in it, the editors of the J. Comp. Phys. obviously do so, and the Rev. too, in almost all cases. But the limiting relation (7.16) (a corrected repetition of the misprinted (6.7)) is not new: with \[ -z\Gamma (-z)\Gamma (z)=-\Gamma (-z)\Gamma (z+1)=\pi \cos ec \pi z \] we find that (7.16) is a special case of an elementary relation on binomial coefficients: \[ \sum^{n}_{k=0}\left( \begin{matrix} a\\ k\end{matrix} \right)\left( \begin{matrix} b\\ n-k\end{matrix} \right)=\left( \begin{matrix} a+b\\ n\end{matrix} \right), \] here \(a=-b=\).