Two matrix inversions associated with the Hagen–Rothe formula, their q-analogues and applications

By means of the Hagen–Rothe formula, we establish two new matrix inversions with four parameters. These new inversions uniformize Riordanʹs inverse relations of Abel-, Chebyshev-, and Legendre-type as well as Gouldʹs inversions based on Vandermonde-type convolutions. Some related q-series inverse relations using the known q-analogues of the Hagen–Rothe formula are established. A Λ-extension of Gouldʹs g-inverse, a novel expression for all Chebyshev-type inversions, and several new summation and transformation formulas of series are presented as applications.