Some Identities and Inequalities Involving Symmetry Sums of Legendre Polynomials

Abstract

By using the analysis methods and the properties of Chebyshev polynomials of the first kind, this paper studies certain symmetry sums of the Legendre polynomials, and gives some new and interesting identities and inequalities for them, thus improving certain existing results.

Ferrers functions of arbitrary degree and order and related functions

“…In particular, P −µ ν (x) arise as a result of the separation of variables in various physical problems, in expansions of functions in series and integrals of Ferrers functions as well as in analytical and numerical approximations based on using orthogonal polynomials or functions. There are numerous publications in this classical field, and recent articles by Bakaleinikov and Silbergleit [4], Bielski [6], Cohl, Dang and Dunster [10], Durand [12], [13], Maier [20], [21], Nemes and Daalhuis [24], Szmytkowski [29], [30], Wang and Qiao [33], and Zhou [35] should be mentioned in this connection. In this article, we obtain for Ferrers functions novel integral representations, which are used, together with analytical continuation, for the systematic derivation of numerous series representations, integral and series connection formulas, asymptotic and differentiation formulas, generating functions and additional theorems for P −µ ν (tanh (α + β)).…”