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 text+ytmλet−1=∑n=0∞Bn(x,y;λ)tnn!,2ext+ytmλet+1=∑n=0∞En(x,y;λ)tnn!.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↦λxBn(x,y;λ),x↦λxBn(x,y;λ),y↦λxBn(x,y;λ)y↦λxBn(x,y;λ) and (x,y)↦λxBn(x,y;λ)(x,y)↦λxBn(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.