On Series Containing Products of Legendre Polynomials

Explicit formulas are established for infinite sums of products of three or four Legendre polynomials of $n$th order with coefficients $2n + 1$; the series depends only the arguments of the polynomials and contains no other variables. We show that, for the product of three polynomials, the sum is inverse to the root of the product of four sine functions and, in the case of four polynomials, this expression additionally contains the elliptic integral $\mathbf{K}(k)$ as a multiplier. Analogs and particular cases are considered which allow one to compare the relationships proved in this note with results proved in various domains of mathematical physics and classical functional analysis.