The infinite sum of a single Legendre Polynomial has a well known expression. Are there any explicit formulas for the infinite sum of the product of two Legendre Polynomials? I'm interested on polynomials of both First and Second kind (usually denoted $P_l$ and $Q_l$ respectively).
$\sum_{l=0}^{\infty} P_l(x)P_l(y)= ?$
$\sum_{l=0}^{\infty} P_l(x)Q_l(y)= ?$
and variants of this, like inserting $(i)^l$.
Some useful properties of Legendre polynomials and its applications to neutron transport equation in slab geometry,by F. Anli and S. Güngör, Appl. Math. Modelling https://www.sciencedirect.com/science/article/pii/S0307904X05002660