A=B by Petkovsek M., Wilf H.S., Zeilberger D.

By Petkovsek M., Wilf H.S., Zeilberger D.

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There is such a wealth of information available now that it is important to have systematic ways of searching the literature for information that may help us to deal with a particular sum. So our main task in this chapter will be to show how a given sum is described by using standardized hypergeometric notation. ” 34 The Hypergeometric Database be much easier to consult databases of known information about such sums. An entry in such a database is a statement to the effect that a certain hypergeometric series is equal to a certain much simpler expression, for all values of the various free parameters that appear, or at least for all values in a suitably restricted range.

N (−1)k k! k tk+1 = tk = k−n , (k + 1)2 and t0 = 1. 1) our unknown sum is revealed to be a 1F1 −n ;1 . 5. Is the Bessel function ∞ Jp (x) = (−1)k ( x2 )2k+p k=0 k! (k + p)! a hypergeometric function? The ratio of consecutive terms is (−1)k+1( x2 )2k+2+pk! (k + p)! tk+1 = tk (k + 1)! (−1)k ( x2 )2k+p 2 −( x4 ) = . (k + 1)(k + p + 1) Here we must take note of the fact that t0 = 1, whereas the standardized hypergeometric series begins with a term equal to 1. Our conclusion is that the Bessel function is indeed hypergeometric, and it is in fact Jp (x) = ( x2 )p x2 ··· F ; − .

SumF, and you will, or should, be looking at the hypergeometric designation of your sum as output. 42 The Hypergeometric Database As an example, take the Laguerre polynomial that we tried in Mathematica. SumF. The output will be the desired hypergeometric form −n ;1 . 5 Some entries in the hypergeometric database The hypergeometric database can be thought of as the collection of all known hypergeometric identities. The following are some of the most useful database entries. We will not prove any of them just now because all of their proofs will follow instantly from the computer certification methods that we will develop in Chapters 4–7.

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