On a transformation of Riordan moment sequences

Abstract

We define a transformation that associates certain exponential moment sequences with ordinary moment sequences in a natural way. The ingredients of this transformation are series reversion, the Sumudu transform (a variant of the Laplace transform), and the inverting of generating functions. This transformation also has a simple interpretation in terms of continued fractions. It associates lattice path objects with permutation objects, and in particular it associates the Narayana triangle with the Eulerian triangle.

Three Études on a sequence transformation pipeline

“…expands to given the Eulerian triangle E 3 that begins At this stage we can invoke the T transform [2] to map E 1 to N 1 and to map E 2 to N 2 . These relationships can also be seen clearly in terms of the Deléham notation.…”

“…, and multiply each on the right by the inverse of the binomial matrix, we obtain respectively the Euler triangle E 1 and the Narayana triangle N 1 . As shown in [2], the two triangles E 1 and N 1 are paired triangles under the T transform.…”

“…In this note, we shall define a transformation pipeline beginning with certain sequences possessing a simple ordinary generating function, that combines the inverse Sumudu transform [9,10,34] (or inverse Laplace Borel transfom) with the reversion of exponential generating functions. Previously, we have studied the invertible T transform, that uses inversion, the Sumudu transform and reversion, beginning with an exponential generating function [2].…”