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

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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 ; − . 0 1 p! p+1 4 ✷ 2 We hope to convince you that a better first step is to reach for your computer! 4 Software that identifies hypergeometric series 39 We will use the notation a(a + 1)(a + 2) · · · (a + n − 1), def (a)n = 1, if n ≥ 1; if n = 0.

Hypergeometric series are very important in mathematics. Many of the familiar functions of analysis are hypergeometric. These include the exponential, logarithmic, trigonometric, binomial, and Bessel functions, along with the classical orthogonal polynomial sequences of Legendre, Chebyshev, Laguerre, Hermite, etc. 3 How to identify a series as hypergeometric 35 metric, then identifying precisely which hypergeometric function it is, and finally by using known results about such functions. 1) where x is a constant.

In the case of proving binomial coefficient identities, the WZ method is a standardized proof procedure that is almost independent of the particular identity that you’re trying to prove. The only thing that changes in the proof, as we go from one 7 Well, hardly ever. 3 Human and computer proofs; an example 25 identity to another, is a certain rational function R(n, k) of two variables, n and k. Otherwise, all of the proofs are the same. So when your computer finds a WZ proof, it doesn’t have to recite the whole thing; it needs to describe only the rational function R(n, k) that applies to the particular identity that you are trying to prove.