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

Abstract

By means of the Hagen-Rothe formula, we establish two new matrix inversions with four parameters. These new inversions uniformize Riordan's inverse relations of Abel-, Chebyshev-, and Legendre-type as well as Gould's 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 Gould's g-inverse, a novel expression for all Chebyshev-type inversions, and several new summation and transformation formulas of series are presented as applications.

Duals of Bernoulli Numbers and Polynomials and Euler Number and Polynomials

“…Krattenthaler defines a class of matrix inverses in [26], which has numerous famous special cases presented by Gould and Hsu [13], Carlitz [4], Bressoud [2], Chu and Hsu [9], Ma [29], etc. Hsu, Shiue, and one of the authors study Riordan matrix inverses in [22].…”

A family of summation formulas on the Fox–Wright function

“…The corresponding results, which can be seen in the papers Chu and Wang (2008); Krattenthaler (1996); Wei et al (2012), are all from inversion techniques and identities related to Saalschutz's theorem. The reader may refer to Ma (2007Ma ( , 2011; Warnaar (2002) for more details on inversion techniques.…”