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

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  • Bayad, Abdelmejid
  • Navas, Luis

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.

Suggested Citation

  • Bayad, Abdelmejid & Navas, Luis, 2015. "Algebraic properties and Fourier expansions of two-dimensional Apostol–Bernoulli and Apostol–Euler polynomials," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 883-892.
    • Handle: RePEc:eee:apmaco:v:265:y:2015:i:c:p:883-892
    DOI: 10.1016/j.amc.2015.05.128
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