Inverse Relations and Schauder Bases

Abstract

The concept of inter-changes of Schauder bases is used to interpret inverse relations for sequences. For a given power series, the interplay between different representations by Schauder bases can result in combinatorial identities, new or known. Local cohomology residues and local duality are used for computations. The viewpoint of Riordan arrays is examined using Schauder bases.

Two Approaches to Möbius Inversion

“…It is shown in [7,Theorem 2.1] that an inverse relation with the orthogonal property comes from representations of a power series by two Schauder bases in the sense of [7]. Assume that all principal filters on X are finite.…”

“…An inversion formula is regarded as a phenomenon of change of 'coordinate systems'. It is shown that Lagrange inversion [5], MacMahon's master theorem [4, Example 1], Saalschütz's theorem [6,Identity 11] and Dixon's theorem [6,Identity 12] are phenomena of changes of variables; an inverse relation is a phenomenon of a change of Schauder bases [7]; Jacobi's formula is a phenomenon of a change of parameters [9]. The purpose of this article is to show that Möbius inversion for a locally finite partially ordered set is within the same view.…”

Commutators and commutator subgroups of the Riordan group

“…The Riordan group was introduced, under this name and in a more restrictive context, by L. Shapiro and collaborators in [35]. Since then, many authors have developed this topic [14,22,27,37,40,42]. Some previous related results can be found in [2,16,17,32,33,39].…”

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

“…Since his work a number of methods for finding matrix inversions have been developed, among which the following are worthwhile mentioning: the integral representation approach (Egorychev [19]); the umbral calculus (Roman [46]); the operator method (Krattenthaler [36]); the local cohomology residues over Schauder-bases (Huang [33]); the Riordan group and array (Shapiro and Sprugnoli [41,42,53,57]); the telescopic approach on formal factorials (Chu [14,13]); the recurrence relation method (Milne and Bhatnagar [43]). Further reference may also extend to [6,7] concerning the U (n)-extensions of inverse relations on root systems, and to [48] as to the theory of rather general Möbius-Rota inversions over partially ordered sets.…”