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PreprintCC BY-NC-SA 4.0arXiv

Novel computational formulas and relations for special numbers and functions: approach to analysis of Debye functions

  • Yilmaz Simsek
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Abstract

The first goal of this paper is to give a new family of special function related to the Debye functions. By using these function and generating function method, we derive many novel formulas and relations involving the Debye functions, multiple Hurwitz Lerch zeta function which interpolate the Apostol-Bernoulli numbers of higher order at negative integers. Proof method of theorems in this paper are different from those of theory of analytic numbers. The second goal of this paper is to show that special values and integral presentations of this function are closely related to unification of the Debye functions, the moment generating function for the negative binom distribution, the generating functions for the Apostol-type numbers of higher-order, and the Frobenius-Euler numbers, the Bernoulli numbers, and the Stirling numbers etc. Moreover, we give recurrence relations, Maclaurin's series expansion, and approximation formula for unification of the Debye functions. Finally, we give further remarks, observations and applications in mathematical physics on the results of this paper.

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Cite

BibTeXRIS
BibTeX
@misc{simsek2026novel,
  title = {{Novel computational formulas and relations for special numbers and functions: approach to analysis of Debye functions}},
  author = {Yilmaz Simsek},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06620},
  eprint = {2609.06620},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06620},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Novel computational formulas and relations for special numbers and functions: approach to analysis of Debye functions
AU  - Yilmaz Simsek
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06620
UR  - https://arxiv.org/abs/2609.06620
AB  - The first goal of this paper is to give a new family of special function related to the Debye functions. By using these function and generating function method, we derive many novel formulas and relations involving the Debye functions, multiple Hurwitz Lerch zeta function which interpolate the Apostol-Bernoulli numbers of higher order at negative integers. Proof method of theorems in this paper are different from those of theory of analytic numbers. The second goal of this paper is to show that special values and integral presentations of this function are closely related to unification of the Debye functions, the moment generating function for the negative binom distribution, the generating functions for the Apostol-type numbers of higher-order, and the Frobenius-Euler numbers, the Bernoulli numbers, and the Stirling numbers etc. Moreover, we give recurrence relations, Maclaurin's series expansion, and approximation formula for unification of the Debye functions. Finally, we give further remarks, observations and applications in mathematical physics on the results of this paper.
ER  -