Algebraic properties and Fourier expansions of two-dimensional Apostol-Bernoulli and Apostol-Euler polynomials
We define the two-dimensional (2D) Apostol-Bernoulli and the 2D Apostol-Euler polynomials respectively via the generating functions t e x t + y t m λ e t - 1 = ? n = 0 ∞ B n ( x , y ; λ ) t n n ! , 2 e x t + y t m λ e t + 1 = ? n = 0 ∞ E n ( x , y ; λ ) t n n ! . As parametrized polynomial families they are essentially the same. We study their basic algebraic properties, generalizing some well-known formulas and relations for Apostol-Bernoulli and Bernoulli polynomials. We determine the Fourier series of x ? λ x B n ( x , y ; λ ) , y ? λ x B n ( x , y ; λ ) and ( x , y ) ? λ x B n ( x , y ; λ ) for (x, y) ? 0, 1) × 0, 1). These contain as a special case the Fourier series of the one-dimensional Apostol-Bernoulli and Apostol-Euler polynomials.