Abstract
In this paper, we study sums of finite products of Legendre and Laguerre polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we are going to express those sums of finite products as linear combinations of Bernoulli polynomials. Further, by using a method other than Fourier series expansions, we will be able to express those sums in terms of Euler polynomials.
Highlights
Introduction and preliminaries TheCoulomb potential can be written as a series in Legendre polynomials Pn(x) (n ≥ 0)
5 Results and discussion In this paper, we investigated sums of finite products of Legendre and Laguerre polynomials and derived Fourier series expansions of functions associated with those polynomials
From these Fourier series expansions, we were able to express those sums of finite products as linear combinations of Bernoulli polynomials
Summary
We start with the following lemma which will play an important role . where the sum is over all nonnegative integers i1, i2, . . . , i2r+1 with i1 + i2 + · · · + i2r+1 = n. Is the Gauss hypergeometric function with x n denoting the rising factorial polynomial defined by x n = x(x + 1) · · · (x + n – 1) (n ≥ 1), x 0 = 1. Combining (2.1) and (2.7), we get the following lemma. Lemma 2.2 For integers n, r with n, r ≥ 0, we have the following identity:. Defined on R, which is periodic with period 1. The Fourier series of αm,r( x ) is. From (1.7), (1.8), (2.10), (2.11), (2.19), and (2.20), we have the following Fourier series expansion of αm,r( x ):. Pi1 x Pi2 x · · · Pi2r+1 x i1 +i2 +···+i2r+1 =m has the Fourier series expansion. M–j+1,r+j–1Bj x j=0 j=1 for all x ∈ R. We observe that the statement in Theorem A follows immediately from Theorems 2.3 and 2.4
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