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

作者：Bayad Abdelmejid; Navas Luis^*

来源：Applied Mathematics and Computation, 2015, 265: 883-892.

DOI：10.1016/j.amc.2015.05.128

• 2D Apostol-Bernoulli polynomials
• Generating functions
• Difference equations
• Fourier expansions

We define the two-dimensional (2D) Apostol-Bernoulli and the 2D Apostol-Euler polynomials respectively via the generating functions
te(xt+ytm)/lambda e(t) - 1 - Sigma B-infinity(n=0)n(x, y; lambda) t(n)/n!, 2e(xt+ytm)/lambda e(t) + 1 - Sigma(infinity)(n=0)epsilon(n)(x, y; lambda) 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 bar right arrow lambda B-x(n)(x, y; lambda), y bar right arrow lambda B-x(n)(x, y; lambda) and (x, y) bar right arrow lambda B-x(n)(x, y; lambda) for (x, y) is an element of [0, 1) x [0, 1). These contain as a special case the Fourier series of the one-dimensional Apostol-Bernoulli and Apostol-Euler polynomials.

• 出版日期2015-8-15