Riordan arrays and the Abel-Gould identity
✍ Scribed by Renzo Sprugnoli
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 1008 KB
- Volume
- 142
- Category
- Article
- ISSN
- 0012-365X
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✦ Synopsis
We generalize the well-known identities of Abel and Gould in the context of Riordan arrays. This allows us to prove analogous formulas for Stirling numbers of both kinds and also for other quantities.
Introduction
Recently, Shapiro et al. [S] have formally introduced the concept of a
Riordan group; it corresponds to a set of infinite, low-triangular arrays characterized by two analytic functions: the first is invertible and the second has a compositional inverse. Even though the concept can be traced back to a paper by Rogers [6] on renewal arrays, the authors give a clear formulation of the theory of Riordan arrays and relate it to the l-umbra1 calculus, as described, for instance, by Roman [7].
We believe that Riordan arrays are particularly important not only theoretically but also because they constitute a practical device for solving combinatorial sums by means of generating functions. These arrays are precisely the class of objects that allow us to translate a sum CiZo dn,kfk into a suitable transformation of the generating functionf(t)=B,{fk)tEN= 9 { f,} of the sequence { fk}fe N. In [9] we tried to give an accurate description of this fact.
Moreover, the concept of Riordan group is strictly related to the Lagrange inversion formula (LIF) which, in turn, is the natural device for inverting the elements in the group. In particular, many traditional applications of the LIF can be approached from a Riordan array point of view. In this paper, we focus our attention on the following two identities of Abel and Gould, respectively:
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