Left-inversion of combinatorial sums
type: Item
instance of: scholarly article
title: Left-inversion of combinatorial sums
zbMATH Open document ID: 0903.05005
DOI: 10.1016/S0012-365X(97)00110-6
author: Cristiano Corsani
author: Donatella Merlini
author: Renzo Sprugnoli
published in: Discrete Mathematics
publication date: 1999-01-05T00:00:00Z
review text: Let \(\kappa\) be any field of characteristic zero; e.g., \(\kappa= \mathbb{R}\) or \(\mathbb{C}\). Consider the algebra \(\kappa t\) of formal series \(f(t)= \sum_{k\geq 0} f_kt^k\). The notation \([t^k] (\dots)\) for `the coefficient of \(t^k\) in the series \((\dots)\)' is being used; so, \([t^k] f(t)= f_k\) and \(f(t)\) serves for the usual generating function of \((f_k\mid k\in \mathbb{N}_0)\). A Riordan array is a pair \((d(t), h(t))\) of formal series with \(d(t)\) of order 0; it defines the triangular array \((d_{n,k} \mid n,k\in \mathbb{N}_0\) and \(k\leq n)\) of elements in \(\kappa\) according to the rule \(d_{n,k}= [t^n]d(t) (th(t))^k\). For such an array \(\sum^n_{k=0} d_{n,k} f_k=[t^n] d(t)f (th(t))\) holds. An example of a Riordan array is provided by \(d(t)= h(t)= (1-t)^{-1}\). It defines the Pascal triangle. Any series \(f(t)\) of order one has the inverse, i.e. the unique series \(g(t)\) such that \((f\circ g)(t)= t=(g \circ f)(t)\). The coefficients of \(g(t)\) are given by Lagrangian inversion, \(g_n= {1\over n!} \Biggl[\Bigl( {d\over dt} \Bigr)^{n-1} \Bigl({t \over f(t)} \Bigr)^n \Biggr]_{t=0}\) \((n\geq 1)\). Inverting combinatorial sums may be considered a center of attraction in combinatorics [see \textit{J. Riordan}, Am. Math. Mon. 71, 485-498 (1964; Zbl 0128.01603) or Combinatorial identities (1968; Zbl 0194.00502)]. In this paper the concepts of Riordan array and Lagrange inversion are used to give a new approach to this question. To explain the essence of what is proposed the authors use the relation \(a_n= \sum^{\lfloor {n\over 2} \rfloor}_{k =0} {n \choose 2k} b_k\) with \((b_k\mid k\in \mathbb{N}_0)\) given. To invert this combinatorial sum, take the array \(D=((1-t)^{-1}\), \(t(1-t)^{-2})\) with \(d_{n,k}= [t^n]((1-t)^{-1} (t(1-t)^{-2})^k) ={n \choose 2k}\). As the diagonal elements \(d_{n,n}\) \((n>0)\) are all zero, it is impossible to use Cauchy inversion. However, the following way out is proposed. For the generating functions…
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A40
zbMATH DE Number: 1135934
zbMATH Keywords: inverting combinatorial sums
zbMATH Keywords: Riordan array
zbMATH Keywords: Lagrangian inversion
reviewed by: U. Kaljulaid
MaRDI profile type: Publication
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Wikidata QID: Q127526473
full work available at URL: s0012-365x(97)00110-6
OpenAlex ID: W2070052785
description: scientific article; zbMATH DE number 1135934