Algebraic properties and Fourier expansions of two-dimensional Apostol–Bernoulli and Apostol–Euler polynomials

Abdelmejid Bayad and Luis Navas

Applied Mathematics and Computation, 2015, vol. 265, issue C, 883-892

Abstract: 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),y↦λxBn(x,y;λ) and (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.

Keywords: 2D Apostol–Bernoulli polynomials; Generating functions; Difference equations; Fourier expansions (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:265:y:2015:i:c:p:883-892

DOI: 10.1016/j.amc.2015.05.128

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