On series containing products of Legendre polynomials (Q881075)

The problem examined in the present paper is an explicit expression for the sum \[ A(x,y,z)=\sum_{n=0}^\infty (2n+1)P_n(x)P_n(y)P_n(z), \quad x,y,z \in (-1,1), \] \(P_n\) are the standard Legendre polynomials. The answer is \[ \pi A(x,y,z)=\left\{ \sin\frac{\lambda-\phi+\psi}2 \sin\frac{\phi-\psi+\lambda}2 \sin\frac{\phi+\psi-\lambda}2 \sin\frac{\phi+\psi+\lambda}2\right\}^{-1/2} \] \(x=\cos\lambda\), \(y=\cos\phi\), \(z=\cos\psi\), in the domain, where the right hand side is positive, and zero otherwise. The similar expression (several different ones) are also obtained for the sum \[ \pi^2 \sum_{n=0}^\infty (2n+1)P_n(x)P_n(y)P_n(z)P_n(t). \] The expressions involve elliptic integrals.