Summability of the Heine and Neumann Series of Legendre Polynomials

Abstract

With a holomorphic function f(z) defined in a domain H which includes the closed interval [—1, 1] we associate the Neumann series1where Pn(z), Qn(t) are, respectively, the nth Legendre polynomials of the first and second kind and γ is a closed and rectifiable Jordan curve which includes [— 1, 1] in its interior and is included, together with its interior, in H.