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Combinatorial identities and inverse binomial coefficients. (English) Zbl 1004.05011

In this paper a general method is presented from which one can obtain a wide class of combinatorial identities. The following is the main result. Let r, nk be any nonnegative integers, and let f(n,k) be given by

f(n,k)=(n+r)! n! u 1 u 2 p k (t)q n-k (t)dt,

where p(t) and q(t) are functions defined on [u 1 ,u 2 ]. Let {a n ,n0} and {b n ,n0} be any two sequences, and let A(x), B(x) be the corresponding ordinary generating functions. Then

n0 k=0 n f (n,k) a k b n-k x n =D r x r u 1 u 2 A [xp(t)] B [xq(t)] d t,

where D r denotes the rth derivative with regard to x.

The above result is generalized to functions represented by integrals over a real d-dimensional domain. Numerous examples illustrating the use of these two results are also given.


MSC:
05A19Combinatorial identities, bijective combinatorics
05A10Combinatorial functions