01 - MAD 6206 Combinatorics I, Fall 2011 CRN: 87401 Brief...

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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)! = (1 z )( z ) = sin( z ) , (1) giving that ( z )! ( 2 z )! = (2 z 1)! sin(2 z ) ( z 1)! sin( z ) = (2 z )! z ! cos( z ) . Thus, if n is a positive integer, then lim z n ( z )!...
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01 - MAD 6206 Combinatorics I, Fall 2011 CRN: 87401 Brief...

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