Estimates of sums of integrals of Legendre polynomial

Abstract

Estimates of sums Rnk(x) = ∞ m=n Pmk(x) are established. Here Pn0(x) = Pn(x), Pnk(x) = x −1 Pn,k−1(y)dy, Pn being a Legendre polynomial with the standard normalisation Pn(1) = 1. If k = 1 then the sum decreases with growing n as n−1 if x belongs to the main segment [−1, 1], wheras it decreases as n−3/2 if x belongs to the semi-segment [−1, 1). If k > 1 the point x = 1 does not need to be excluded. The sum decreases as n−k−1/2. The more, a small increasing of the multiplicative constant permits to obtain an estimate |Rnk(cos θ)| < C sink−3/2 θ nk+1/2, where C depends weakly on k (but not on n, θ). As a by-product, the integral of Mehler—Dirichlet type for Rnk(cos θ) is deduced. Refs 6. Figs 3. Tables 1.