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PreprintCC BY-NC-ND 4.0arXiv

Certain families of series involving Hurwitz and Lerch zeta functions

  • Dora Pokaz
  • Mihaela Ribicic Penava
  • Yilmaz Simsek
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Abstract

This paper deals with a certain family of series which is derived from generating functions of the Frobenius-Euler and Apostol type polynomials and numbers. When special cases of this series are examined, it is shown that it generates some well-known families of zeta functions, such as the Hurwitz zeta and the Lerch zeta, etc. We also give an open problem for these relation. We give analytic continuation of this function. By using the Cauchy Residue Theorem, We show that this series interpolates the Frobenius-Euler type polynomials at negative integers. Moreover, by applying the Laplace transform to the generating function for these polynomials, we get infinite series representations involving the Lerch zeta function and these polynomials. In addition, We show that this interpolation function can be representation in terms of the Lerch zeta function and the Hurwitz zeta function. Some special values for the results are also given.

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Cite

BibTeXRIS
BibTeX
@misc{pokaz2026certain,
  title = {{Certain families of series involving Hurwitz and Lerch zeta functions}},
  author = {Dora Pokaz and Mihaela Ribicic Penava and Yilmaz Simsek},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13796},
  eprint = {2609.13796},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13796},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Certain families of series involving Hurwitz and Lerch zeta functions
AU  - Dora Pokaz
AU  - Mihaela Ribicic Penava
AU  - Yilmaz Simsek
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.13796
UR  - https://arxiv.org/abs/2609.13796
AB  - This paper deals with a certain family of series which is derived from generating functions of the Frobenius-Euler and Apostol type polynomials and numbers. When special cases of this series are examined, it is shown that it generates some well-known families of zeta functions, such as the Hurwitz zeta and the Lerch zeta, etc. We also give an open problem for these relation. We give analytic continuation of this function. By using the Cauchy Residue Theorem, We show that this series interpolates the Frobenius-Euler type polynomials at negative integers. Moreover, by applying the Laplace transform to the generating function for these polynomials, we get infinite series representations involving the Lerch zeta function and these polynomials. In addition, We show that this interpolation function can be representation in terms of the Lerch zeta function and the Hurwitz zeta function. Some special values for the results are also given.
ER  -