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

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Here a good tactic would be to insert an extra xk into the sum, hope that 40 The Hypergeometric Database Mathematica does not know any simple form for that one, and then put x = 1 in the answer. So we hopefully type Sum[Binomial[n, k]ˆ2 xˆk, {k, 0, n}], and sure enough it responds with Hypergeometric2F1[−n, −n, 1, x], and now we can let x = 1 to learn that our original sum was a 2F1 −n, −n ;1 . 4 above. , {k, 0, n}], we find that Mathematica is very well trained indeed, since it gives LaguerreL[n, 0, 1] which means that it recognizes our sum as a Laguerre polynomial!

Chapter 5 is about the fundamental algorithm of Gosper, which is to summation as finding antiderivatives is to integration. This algorithm allows us to do indefinite hypergeometric sums in simple closed form, or it furnishes a proof of impossibility if, in a given case, that cannot be done. Beyond its obvious use in doing indefinite sums, it has several nonobvious uses in executing the WZ method, in finding recurrences for definite sums, and even for finding the right hand side of a definite sum whose evaluation we are seeking.

Computers can help even with this humble task. In this section we’ll discuss the use of Mathematica, Maple, and a special purpose package Hyp that was developed by C. Krattenthaler. First, Mathematica has a limited capability for transforming sums that are given in customary summation form into standard hypergeometric form. This capability resides in the package Algebra‘SymbolicSum‘. So we first read in the package, with <

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