AI 中文总结
本文定义系数为k增中心三角数的可逆Hurwitz-Lerch型函数，推导其收敛条件、反演公式等，所得结果可还原经典Hurwitz-Lerch超越函数及其导数，还给出相关递推公式与特殊值表达。
AI 中文摘要
本文定义了一类系数为k增中心三角数的Hurwitz-Lerch型函数，针对该类函数，我们得到了收敛条件、约化公式及欧拉算子形式；针对一类带多项式权重的Hurwitz-Lerch函数，推导了基于范德蒙德的反演公式。所研究的类为几何因子1、2、4的二次情形，所得公式表明，三个连续函数可还原经典Hurwitz-Lerch超越函数及其前两个欧拉导数。我们还推导了递推公式、普通生成函数、有限和及特殊值，z=1处的值通过Hurwitz ζ函数和伯努利多项式表示；当a=1时，有理值H_k(z,-m,1)的分子多项式用欧拉多项式表示，而交替值H_k(-1,-m,a)则通过欧拉多项式表示。
英文摘要
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.
Comments17 pages
