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.