Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

Bayad, Abdelmejid

doi:10.1090/s0025-5718-2011-02476-2

Cited by 31 publications

(32 citation statements)

References 4 publications

(2 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The results obtained generalize those in [2,6,8]. These are interesting results both from the theoretical and computational point of view, as they allow the use of Fourier approximation to both calculate and derive further relations between the various polynomial families involved.…”

Section: Introductionsupporting

confidence: 69%

“…As a special case of Theorem 3.1 we obtain the following result, which was first proved in [2]. As for pointwise convergence of the Fourier series in the one-dimensional case, Proposition 3.2 specializes to the following result.…”

Section: The One-dimensional Casementioning

confidence: 71%

“…, (2) which converges for |t + log λ| < π . They specialize to the 1D Apostol-Euler and classical Euler polynomials.…”

Section: Introductionmentioning

confidence: 90%

See 2 more Smart Citations

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

Bayad

Navas

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

Section: Introductionsupporting

confidence: 69%

Section: The One-dimensional Casementioning

confidence: 71%

See 1 more Smart Citation

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

Bayad

Navas

2015

Applied Mathematics and Computation

View full text Add to dashboard Cite

“…Then we obtain 2) where the inner sum is taken over all nonnegative integers i, j, k, l with i + j + k + l = n, and…”

Section: Symmetric Identities From (B-1)mentioning

confidence: 99%

“…Luo and Srivastava [21] generalized this definition to obtain the generalized ApostolBernoulli and Euler polynomials, and they also studied them systematically. Recently, Luo [20], Bayad [2], Navas, Francisco and Varona [23] investigated Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials. Kim and Hu [12] obtained the sums of product identity for the Apostol-Bernoulli numbers which is an analogue of the classical sums of product identity for the Bernoulli numbers dating back to Euler.…”

Section: Introductionmentioning

confidence: 99%

IDENTITIES OF SYMMETRY FOR THE HIGHER ORDER q-BERNOULLI POLYNOMIALS

Son¹

2014

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. The study of the identities of symmetry for the Bernoulli polynomials arises from the study of Gauss's multiplication formula for the gamma function. There are many works in this direction. In the sense of p-adic analysis, the q-Bernoulli polynomials are natural extensions of the Bernoulli and Apostol-Bernoulli polynomials (see the introduction of this paper). By using the N -fold iterated Volkenborn integral, we derive serval identities of symmetry related to the q-extension power sums and the higher order q-Bernoulli polynomials. Many previous results are special cases of the results presented in this paper, including Tuenter's classical results on the symmetry relation between the power sum polynomials and the Bernoulli numbers in [A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001), no. 3, 258-261] and D. S. Kim's eight basic identities of symmetry in three variables related to the q-analogue power sums and the q-Bernoulli polynomials in [Identities of symmetry for q-Bernoulli polynomials, Comput. Math.

show abstract

Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters

Kucukoglu

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

The main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p‐adic q‐integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe‐type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism.

show abstract

Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

Abstract: Abstract. We find Fourier expansions of Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. We give a very simple proof of them.

Cited by 31 publications

References 4 publications

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

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

IDENTITIES OF SYMMETRY FOR THE HIGHER ORDER q-BERNOULLI POLYNOMIALS

Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters

Contact Info

Product

Resources

About