Sums of finite products of Legendre and Laguerre polynomials

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.

[1]  C. Cesarano,et al.  Bernstein polynomials and operational methods , 2006 .

[2]  R. Pandharipande,et al.  Hodge integrals and Gromov-Witten theory , 1998 .

[3]  C. Cesarano,et al.  THE LEGENDRE POLYNOMIALS AS A BASIS FOR BESSEL FUNCTIONS , 2016 .

[4]  Hiroo Miki,et al.  A relation between Bernoulli numbers , 1978 .

[5]  P. E. Ricci,et al.  On a family of hybrid polynomials , 2004 .

[6]  Luis M. Navas,et al.  Algebraic properties and Fourier expansions of two-dimensional Apostol-Bernoulli and Apostol-Euler polynomials , 2015, Appl. Math. Comput..

[7]  Gerald V. Dunne,et al.  Bernoulli number identities from quantum field theory and topological string theory , 2013 .

[8]  Taekyun Kim,et al.  Representing Sums of Finite Products of Chebyshev Polynomials of Third and Fourth Kinds by Chebyshev Polynomials , 2018, Symmetry.

[9]  Ira M. Gessel,et al.  On Miki's identity for Bernoulli numbers , 2005 .

[10]  Katsumi Shiratani,et al.  AN APPLICATION OF p-ADIC CONVOLUTIONS , 1982 .

[11]  Roderick Wong,et al.  Special Functions: Frontmatter , 2010 .

[12]  Taekyun Kim,et al.  Fourier series of finite products of Bernoulli and Genocchi functions , 2017, Journal of inequalities and applications.

[13]  Hari M. Srivastava,et al.  The Laguerre and Legendre polynomials from an operational point of view , 2001, Appl. Math. Comput..

[14]  R. Beals,et al.  Special Functions: Discrete orthogonal polynomials , 2010 .

[15]  G. Andrews,et al.  Special Functions: The Hypergeometric Functions , 1999 .

[16]  Taekyun Kim,et al.  Sums of finite products of Genocchi functions , 2017 .

[17]  Roderick Wong,et al.  Special Functions by Richard Beals , 2010 .

[18]  Yilmaz Simsek,et al.  Formulas for Poisson–Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach , 2019 .

[19]  Taekyun Kim,et al.  Sums of Finite Products of Euler Functions , 2017 .

[20]  Ravi P. Agarwal,et al.  Sums of finite products of Bernoulli functions , 2017 .

[21]  Yilmaz Simsek.  Combinatorial Identities Associated with Bernstein Type Basis Functions , 2016 .

[22]  Taekyun Kim,et al.  Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials , 2018, Journal of inequalities and applications.