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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 $r$, $n\ge k$ 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 $\{a_n, n\ge 0\}$ and $\{b_n, n\ge 0\}$ be any two sequences, and let $A(x)$, $B(x)$ be the corresponding ordinary generating functions. Then $$\sum_{n\le 0} \Biggl[\sum^n_{k=0} f(n,k) a_k b_{n- k}\Biggr] x^n= D^r\Biggl[x^r \int^{u_2}_{u_1} A[xp(t)] B[xq(t)]\,dt\Biggr],$$ 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.

05A19Combinatorial identities, bijective combinatorics
05A10Combinatorial functions
11B65Binomial coefficients, etc.
Full Text: DOI
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