By Petkovsek M., Wilf H.S., Zeilberger D.
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Extra resources for A=B
We have also seen a list of many of the important hypergeometric sums that can be expressed in simple, closed form. We will now give a few examples of the whole process whereby one uses the hypergeometric database in order to try to “do” a given sum. The strengths and the limitations of the procedure should then be clearer. 1. For a nonnegative integer n, consider the sum (−1)k f (n) = k 2n . k Can this sum be evaluated in some simple form? The first step is to identify the sum f(n) as a particular hypergeometric series.
Precisely when can we expect a pretty answer? How fast is it? What is the complete algorithm, including the simplifications at the end, and how costly are they? What are the alternatives? In the sequel we will present other algorithms, ones that don’t involve any lookup in or manipulation of a database, for doing hypergeometric sums in simple form. Those algorithms can be rather easily programmed for a computer, they work under conditions that are wider than those of the database lookup, and the conditions under which they work can be clearly stated.
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.
A=B by Petkovsek M., Wilf H.S., Zeilberger D.