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