combinatorics

An Invertible Family of Hurwitz--Lerch Type Functions Associated with -Augmented Centered Triangular Numbers

arXiv:2607.26403

summary

The paper introduces a family of Hurwitz–Lerch type functions whose coefficients are k‑augmented centered triangular numbers, and derives convergence conditions, reduction and inversion formulas, generating functions, and special value expressions involving Hurwitz zeta, Bernoulli, Eulerian, and Euler polynomials.

Abstract

This paper defines a family of Hurwitz--Lerch type functions whose coefficients are the \(k\)-augmented centered triangular numbers. For this family, we obtain the convergence conditions, a reduction formula, and an Euler-operator form. A Vandermonde-based inversion formula is derived for a class of polynomially weighted Hurwitz--Lerch functions. The family considered here is the quadratic case with geometric factors \(1\), \(2\), and \(4\). The resulting formulas show that three consecutive functions recover the classical Hurwitz--Lerch transcendent and its first two Euler derivatives. We also derive recurrence formulas, ordinary generating functions, finite sums, and special values. The values at \(z=1\) are expressed through Hurwitz zeta functions and Bernoulli polynomials. When \(a=1\), the numerator polynomials of the rational values \(H_k(z,-m,1)\) are written in terms of Eulerian polynomials, while the alternating values \(H_k(-1,-m,a)\) are expressed through Euler polynomials.

17 pages

Topics & keywords

#hurwitz‑lerch functions#centered triangular numbers#generating functions#inversion formulas#special valuesHurwitz‑Lerch transcendentk‑augmented centered triangular numbersVandermonde inversionEuler operatorHurwitz zetaBernoulli polynomialsEulerian polynomialsEuler polynomials