A fresh approach to classical Eisenstein series and the newer Hilbert–Eisenstein series

DOI10.1142/S1793042117500464zbMath1422.11039arXiv1604.03245OpenAlexW2962795362MaRDI QIDQ2978904

Tibor K. Pogány, Paul L. Butzer

Publication date: 2 May 2017

Published in: International Journal of Number Theory (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1604.03245

zbMATH Keywords

Riemann zeta functionHilbert transformEisenstein seriesBernoulli numbersexponential generating functionsdigamma functionHilbert-Eisenstein seriesDirichlet eta functionButzer-Flocke-Hauss complete omega functionconjugate Bernoulli numbers

Mathematics Subject Classification ID

Bernoulli and Euler numbers and polynomials (11B68) Integral methods for summability (40C10) Gamma, beta and polygamma functions (33B15) Other functions defined by series and integrals (33E20) Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas) (11M36)

Related Items

A retrospective on research visits of Paul Butzer's Aachen research group to Eastern Europe and Tenerife ⋮ Extension of Mathieu series and alternating Mathieu series involving the Neumann function \(Y_{\nu}\) ⋮ Multi-parameter Mathieu, and alternating Mathieu series ⋮ Generalization of the quotient mean series

Cites Work

• Mathieu series and associated sums involving the zeta functions
• Partial fraction expansion of the cotangens, Herglotz trick, and the Weierstrass continuous, nowhere differentiable function
• A linear ODE for the Omega function associated with the Euler function \(E_{\alpha}(z)\) and the Bernoulli function \(B_{\alpha }(z)\)
• Some families of Mathieu \(\mathbf a\)-series and alternating Mathieu \(\mathbf a\)-series
• The classical and approximate sampling theorems and their equivalence for entire functions of exponential type
• Ramanujan's Place in the World of Mathematics
• Alternating Mathieu Series, Hilbert–Eisenstein Series and Their Generalized Omega Functions
• Some two-sided bounding inequalities for the Butzer-Flocke-Hauss omega function
• Unbestimmte Integrale
• Proofs from THE BOOK
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item