Estimates of sums of integrals of the Legendre Polynomial

Abstract

Estimates of sums \({R_{nk}}\left( x \right) = \sum\limits_{m = n}^\infty {{P_{mk}}\left( x \right)} \) are established. Here, P_n0(x)= P_n(x), \({R_{nk}}\left( x \right) = \int\limits_.^x {{P_{n,k - 1}}\left( y \right)dy} \), P_n is the Legendre polynomial with standard normalization P_n(1) = 1. With k = 1 in the main interval [–1, 1] the sum decreases with increasing n as n^–1, and in the half-open interval [–1, 1), as n^–3/2. With k > 1 the point x = 1 does not need to be excluded. The sum decreases as n^-k–1/2. Moreover, a small increase in the multiplicative constant permits to obtain the estimate \(|{R_{nk}}\left( {\cos \theta } \right)| < \frac{{C{{\sin }^{k - 3/2}}\theta }}{{{n^{k + 1/2}}}}\), where C depends weakly on k (but not on n, θ). In passing, a Mehler–Dirichlet-type integral for R_nk(cos θ) is deduced.