A new family of positive integers
نویسنده
چکیده
Let n, p, k be three positive integers. We prove that the numbers n k 3 F 2 (1 − k, −p, p−n ; 1, 1−n ; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application. z k k! , where for an indeterminate a and some positive integer k, the raising factorial is defined by (a) k = a(a + 1). .. (a + k − 1). There are not many families of positive integers which may be defined in terms of hypergeometric functions. Among them stand of course the binomial coefficients n p = 2 F 1 −p, p − n 1 ; 1. Actually this expression is obtained by specializing x = p − n, y = 1 in the celebrated Chu-Vandermonde formula (y − x) p (y) p = 2 F 1 −p, x y ; 1. Of course using this relation as a definition of binomial coefficients would be rather tautological. However, quite surprisingly, it is possible to define a new family of positive integers by slightly modifying the Chu-Vandermonde formula.
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