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Unformatted text preview: MAD 6206 Combinatorics I, Fall 2011 CRN: 87401 Brief notes/example on hypergeometric series 1 Definition Consider the series ∑ k ≥ t k where t k can depend on some other parameters (and such parameters are regarded as constants). If t k +1 t k = x, for some constant x, then ∑ k ≥ t k is a geometric series . If t k +1 t k = P ( k ) Q ( k ) , for some polynomials P,Q, then ∑ k ≥ t k is a hypergeometric series . Example 1. Consider the series summationdisplay k parenleftbigg n k parenrightbiggparenleftbigg n + a k parenrightbiggparenleftbigg n − a n − k parenrightbigg . Then t k +1 t k = n − k k + 1 n + a − k k + 1 n − k k + 1 − a . Since t k +1 t k is a rational function in k , the sum is therefore hypergeometric. Also, we shall need the formula ( − z )!( z − 1)! = Γ(1 − z )Γ( z ) = π sin( πz ) , (1) giving that ( − z )! ( − 2 z )! = (2 z − 1)! sin(2 πz ) ( z − 1)! sin( πz ) = (2 z )! z ! cos( πz ) . Thus, if n is a positive integer, then lim z → n ( − z )!...
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This note was uploaded on 01/02/2012 for the course MAD 6206 taught by Professor Suen during the Spring '11 term at University of South Florida.
- Spring '11