Absolutely monotonic functions and connection coefficients for polynomials
type: Item
instance of: scholarly article
title: Absolutely monotonic functions and connection coefficients for polynomials
zbMATH Open document ID: 0648.26015
DOI: 10.1016/0022-247X(88)90222-3
author: G. G. Bilodeau
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: The author has shown [same Journal 116, 489-496 (1986; Zbl 0594.26013)] that if \(\{a_ n\}\) is nondecreasing and \((D+a_ n)^ nf(x)\geq 0\) on [a,b] with \(D\equiv d/dx\), then f is analytic on [a,b) if \(\limsup (a_ n/n)<\infty.\) Here he extends this result to a larger class of polynomial differential operators. Much use is made of the connection coefficients that express \((x+a_ n)^ n\) in terms of a given \(\{P_ n(x)\}\), where \(\{a_ n\}\) is nondecreasing. In particular, the quoted theorem persists when \(P_ n(D)f(x)\geq 0,\) provided that the connection coefficients are nonnegative. Applications are made to the cases when \(\{P_ n\}\) is an Appell or Sheffer set, or when \(\{P_ n\}\) satisfies a three-term recurrence relation of the kind satisfied by orthogonal polynomials. An interesting auxiliary result gives necessary and sufficient conditions for an Appell sequence to have all zeros real.
Mathematics Subject Classification ID: 26E05
Mathematics Subject Classification ID: 26A48
zbMATH DE Number: 4057849
zbMATH Keywords: absolutely monotonic functions
zbMATH Keywords: Appell set
zbMATH Keywords: polynomial differential operators
zbMATH Keywords: connection coefficients
zbMATH Keywords: Sheffer set
zbMATH Keywords: Appell sequence
MaRDI profile type: Publication
cites work: Q5581815
cites work: Extensions of Bernstein's theorem on absolutely monotonic functions
cites work: Q3250721
cites work: Q5552079
cites work: Generalized Appell connection sequences
cites work: Q5847356
cites work: Q5849166
cites work: Q4753589
cites work: Q5675806
cites work: What is the Laplace Transform?
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description: scientific article; zbMATH DE number 4057849