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.