General eulerian polynomials as moments using exponential riordan arrays

Paul Barry

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)

Abstract

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the general Eulerian polynomials, as defined by Xiong, Tsao and Hall, are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of this polynomial sequence in terms of continued fractions, and we also calculate the Hankel transforms of the polynomial sequence. We indicate that the polynomial sequence can be characterized by the further notion of generalized Eulerian distribution first introduced by Morisita. We finish with examples of related Pascal-like triangles.

Original languageEnglish
Pages (from-to)1-15
Number of pages15
JournalJournal of Integer Sequences
Volume16
Issue number9
Publication statusPublished - 2013

Keywords

  • Eulerian number
  • Eulerian polynomial
  • Euler’s triangle
  • Exponential Riordan array
  • General Eulerian number
  • Generalized Eulerian polynomial
  • Hankel transform
  • Moment
  • Orthogonal polynomial

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