Harmonic numbers and Lerch functions behind falling factorial generating functions
Abstract
We study a family of special numbers and polynomials introduced by Simsek whose exponential generating function has as denominator a falling factorial in the exponential variable. Using the partial fraction decomposition of this denominator, each member of the family is expressed by a single finite sum involving Bernoulli polynomials, Apostol Bernoulli polynomials and two variable Fubini polynomials evaluated at reciprocals of integers. A second expansion, around the point where the denominator vanishes, expresses the same numbers through Stirling numbers of the second kind and generalized harmonic numbers, and yields closed formulas for the first few members. We construct a function, built from the Hurwitz zeta function and Lerch s transcendent, which interpolates the polynomials at the negative integers and gives a closed form of a multiple series recently introduced by Pokaz, Ribicic Penava and Simsek, we give an Euler type integral representation for it, and show that the numbers are signed combinations of moments of geometric random variables. Recurrence relations, asymptotics and an extension to arbitrary distinct nodes are also given.
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BibTeX
@misc{dutta2026harmonic,
title = {{Harmonic numbers and Lerch functions behind falling factorial generating functions}},
author = {Hemen Dutta and Gradimir V. Milovanovic and Yilmaz Simsek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18893},
eprint = {2610.18893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18893},
}
RIS
TY - UNPB
TI - Harmonic numbers and Lerch functions behind falling factorial generating functions
AU - Hemen Dutta
AU - Gradimir V. Milovanovic
AU - Yilmaz Simsek
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.18893
UR - https://arxiv.org/abs/2610.18893
AB - We study a family of special numbers and polynomials introduced by Simsek whose exponential generating function has as denominator a falling factorial in the exponential variable. Using the partial fraction decomposition of this denominator, each member of the family is expressed by a single finite sum involving Bernoulli polynomials, Apostol Bernoulli polynomials and two variable Fubini polynomials evaluated at reciprocals of integers. A second expansion, around the point where the denominator vanishes, expresses the same numbers through Stirling numbers of the second kind and generalized harmonic numbers, and yields closed formulas for the first few members. We construct a function, built from the Hurwitz zeta function and Lerch s transcendent, which interpolates the polynomials at the negative integers and gives a closed form of a multiple series recently introduced by Pokaz, Ribicic Penava and Simsek, we give an Euler type integral representation for it, and show that the numbers are signed combinations of moments of geometric random variables. Recurrence relations, asymptotics and an extension to arbitrary distinct nodes are also given.
ER -