The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind
type: Item
instance of: scholarly article
title: The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind
zbMATH Open document ID: 0804.33001
DOI: 10.1016/0377-0427(94)00010-7
author: K. S. Kölbig
published in: Journal of Computational and Applied Mathematics
publication date: 1994-09-25T00:00:00Z
review text: The \(p\)-th exponential complete Bell polynomial \(Y_ p (x_ 1,\dots, x_ p)\) is defined by \[ \exp \biggl( \sum_{j=1}^ \infty x_ j {{t^ j} \over {j!}} \biggr)= 1+ \sum_{p=1}^ \infty Y_ p (x_ 1,\dots, x_ p) {{t^ p} \over {p!}}. \] Such polynomial for certain arguments, given essentially by finite sums of reciprocal powers with a real parameter, is expressed in terms of the coefficients \(S(p, q; \alpha)\) generated by the power series expansion of a Pochhammer symbol, i.e. \[ (1+\alpha+ x)_ q= \sum_{p=0}^ q S(p,q; \alpha) x^ p. \] This result generalizes an earlier representation, in terms of Stirling numbers, given by \textit{L. Comtet} [The art of finite and infinite expansions. (1974; Zbl 0283.05001)].
reviewed by: Luigi Gatteschi
Mathematics Subject Classification ID: 33B99
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 641403
zbMATH Keywords: Pochhammer symbol
zbMATH Keywords: Bell polynomial
zbMATH Keywords: Stirling numbers
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)00010-7
OpenAlex ID: W1967244805
cites work: Q3331506
cites work: On the Integral � ∞ 0 e -μt t t ν- 1 log m t dt
cites work: On the integral \(\int ^{\infty}_{0}x^{\nu -1}(1+\beta x)^{- \lambda}\,\ln ^ mx\,dx\)
cites work: Q3243274
description: scientific article; zbMATH DE number 641403