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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 \(r\), \(n\geq k\) be any nonnegative integers, and let \(f(n,k)\) be given by \[ f(n,k)= {(n+ r)!\over n!} \int^{u_2}_{u_1} p^k(t) q^{n-k}(t)\, dt, \] where \(p(t)\) and \(q(t)\) are functions defined on \([u_1,u_2]\). Let \(\{a_n, n\geq 0\}\) and \(\{b_n, n\geq 0\}\) be any two sequences, and let \(A(x)\), \(B(x)\) be the corresponding ordinary generating functions. Then \[ \sum_{n\leq 0} \Biggl[\sum^n_{k=0} f(n,k) a_k b_{n- k}\Biggr] x^n= D^r\Biggl[x^r \int^{u_2}_{u_1} A[xp(t)] B[xq(t)]\,dt\Biggr], \] where \(D^r\) denotes the \(r\)th 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:

05A19 Combinatorial identities, bijective combinatorics
05A10 Factorials, binomial coefficients, combinatorial functions
11B65 Binomial coefficients; factorials; \(q\)-identities
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References:

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