학술논문
The umbral calculus and the solution to certain recurrence relations.
Document Type
Journal
Author
Roman, Steven (1-CAS3) AMS Author Profile; De Land, Paul (1-CAS3) AMS Author Profile; Shiflett, Ray (1-CAS3) AMS Author Profile; Shultz, Harris (1-CAS3) AMS Author Profile
Source
Subject
05 Combinatorics -- 05A Enumerative combinatorics
05A40Umbral calculus
05A40
Language
English
Abstract
Introduction: ``We show how the modern umbral calculus can be used to solve certain recurrence relations for Sheffer sequence solutions. One of the simplest nontrivial examples is that of the three-term recurrence $s_{n+1}(x)=(x-b_n)s_n(x)-d_ns_{n-1}(x)$ known to characterize polynomial sequences orthogonal with respect to a moment functional. The Sheffer sequence solutions to this recurrence were determined in 1934 by Meixner. Our approach gives a very elementary solution to this and other recurrence relations.''