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By Marko Petkovsek, Herbert S. Wilf

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2m s (n + 1)n−1 = 1+ n≥1  = (−1)m (3m)! 2) tn , n! 4) Beautiful identities have often stimulated mathematicians to find correspondingly beautiful proofs for them; proofs that have perhaps illuminated the combinatorial or other significance of the equality of the two members, or possibly just dazzled us with their unexpected compactness and elegance. It is a fun activity for people to try to prove identities. We have been accused of taking the fun out of it by developing these computer methods6 but we hope that we have in fact only moved the fun to a different level.

Is thereby defined for all complex numbers n other than the negative integers. Some of the relationships between these three notations are Γ(n + 1) = n! = (1)n , (a)n = a(a + 1) · · · (a + n − 1) = (a + n − 1)! Γ(n + a) = . (a − 1)! Γ(a) (I) Gauss’s 2 F1 identity. If b is a nonpositive integer or c − a − b has positive real part, then Γ(c − a − b)Γ(c) a b ;1 = . 2F1 c Γ(c − a)Γ(c − b) (II) Kummer’s 2 F1 identity. If a − b + c = 1, then Γ( 2b + 1)Γ(b − a + 1) a b ; −1 = . 5 Some entries in the hypergeometric database 43 If b is a negative integer, then this identity should be used in the form 2F1 πb Γ(|b|)Γ(b − a + 1) a b ; −1 = 2 cos ( ) |b| , c 2 Γ( 2 )Γ( 2b − a + 1) which follows from the first form by using the reflection formula π Γ(z)Γ(1 − z) = sin πz for the Γ-function and taking the limit as b approaches a negative integer.

If there is no factor of k! in the denominator of your term tk , just put it in, and compensate for having done so by putting an extra factor in the numerator. 3 How to identify a series as hypergeometric Many of the famous functions of classical analysis have hypergeometric series expansions. 1). − So ex = 0F0 ; x . − What we want to do now is to show how one may identify a given hypergeometric series as a particular p Fq[· · · ]. This process is at the heart of using the hypergeometric 36 The Hypergeometric Database database.

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A=B by Marko Petkovsek, Herbert S. Wilf

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