Estimates of sums of integrals of Legendre polynomial


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.Работа выполнена при финансовой поддержке РФФИ (грант 14-02-00804) и СПбГУ (грант 6.37.341.2015)

Similar works

Full text


Saint Petersburg State University

Full text is not available time updated on 7/9/2019

This paper was published in Saint Petersburg State University.

Having an issue?

Is data on this page outdated, violates copyrights or anything else? Report the problem now and we will take corresponding actions after reviewing your request.