Riordan arrays and the Abel-Gould identity. (English) Zbl 0832.05007

Sprugnoli, Renzo

Discrete Math. 142, No. 1-3, 213-233 (1995).

Riordan arrays were introduced in [L. V. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Appl. Math. 34, No. 1-3, 229-239 (1991; Zbl 0754.05010)] and further developed by the author of the present paper in a previous work [the author, Riordan arrays and combinatorial sums, Discrete Math. 132, No. 1- 3, 267-290 (1994; Zbl 0814.05003)]. Here the author uses the Lagrange inversion formula and his previous results to obtain a general formula from which the so-called Abel-Gould identity and many other identities, including some involving Stirling numbers of both kinds, fall out as special cases.

Reviewer: T.R.Walsh (Montreal)

MSC:

05A19	Combinatorial identities, bijective combinatorics
05A15	Exact enumeration problems, generating functions

Keywords:

generating functions; Riordan arrays; Lagrange inversion formula; Abel- Gould identity; Stirling numbers

References:

[1]	Comtet, L., Advanced combinatorics, (1974), Reidel Dordrecht, NL
[2]	Egorychev, G.P., Integral representation and the computation of combinatorial sums, Amer. math. soc. translations, Vol. 59, (1984) · Zbl 0524.05001
[3]	Graham, R.L.; Knuth, D.E.; Patashnik, O., Concrete mathematics, (1988), Addison-Wesley Reading, MA
[4]	Goulden, I.P.; Jackson, D.M., Combinatorial enumeration, (1983), Wiley New York · Zbl 0519.05001
[5]	Riordan, J., Combinatorial identities, (1968), Wiley New York · Zbl 0194.00502
[6]	Rogers, D.G., Pascal triangles, (), 301-310 · Zbl 0398.05007
[7]	Roman, S., The umbral calculations, (1984), Academic Press New York
[8]	Shapiro, L.V.; Getu, S.; Woan, W.-J.; Woodson, L., The Riordan group, Discrete appl. math., 34, 229-239, (1991) · Zbl 0754.05010
[9]	Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete math., 132, 267-290, (1994) · Zbl 0814.05003