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.