Sums of finite products of Legendre and Laguerre polynomials

The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions 1 F 1 and 2 F 1 .

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

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

[3]  Clemente Cesarano,et al.  A note on two-variable Chebyshev polynomials , 2017 .

[4]  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.

[5]  Serkan Araci,et al.  A new class of Laguerre-based Apostol type polynomials , 2016 .

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

[7]  Taekyun Kim,et al.  Expressing Sums of Finite Products of Chebyshev Polynomials of the Second Kind and of Fibonacci Polynomials by Several Orthogonal Polynomials , 2018, Mathematics.

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

[9]  Taekyun Kim,et al.  Sums of finite products of Legendre and Laguerre polynomials , 2018, Advances in Difference Equations.

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

[11]  Taekyun Kim,et al.  Some Identities on Laguerre Polynomials in Connection with Bernoulli and Euler Numbers , 2012 .

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

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

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

[15]  Taekyun Kim,et al.  Some Identities on Bernoulli and Hermite Polynomials Associated with Jacobi Polynomials , 2012 .

[16]  Taekyun Kim,et al.  Some identities involving Gegenbauer polynomials , 2012, 1207.7165.

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

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

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

[21]  Serkan Araci,et al.  The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials , 2013, 1312.7838.

[22]  Taekyun Kim,et al.  Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Numbers and Polynomials , 2012 .

[23]  Taekyun Kim,et al.  Hermite Polynomials and their Applications Associated withBernoulli and Euler Numbers , 2012 .

[24]  Clemente Cesarano,et al.  A NOTE ON A SPECIAL CLASS OF HERMITE POLYNOMIALS , 2015 .

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

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

[27]  Taekyun Kim,et al.  Some identities on Bernoulli and Euler polynomials arising from orthogonality of Legendre polynomials , 2012 .

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

[29]  George E. Andrews,et al.  Special Functions: Partitions , 1999 .

[30]  Ioana Dumitriu,et al.  Special functions and orthogonal polynomials , 2011 .

[31]  Frits Beukers,et al.  SPECIAL FUNCTIONS (Encyclopedia of Mathematics and its Applications 71) , 2001 .

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

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

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