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

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We will see here how to do that, and give some examples of the treasures that can be found in this way. In Chapter 8 we deal with the question of solving linear recurrences with polynomial coefficients. In all of the examples earlier in the book, the computer analysis of an identity will produce just such a recurrence that the sum satisfies. If we want to prove that a certain right hand side is the correct one, then we just check that the claimed right hand side satisfies the same recurrence and we check a few initial conditions.

In this case we will call the terms hypergeometric terms. /((k + 3)! (2k + 7)). 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.

N − 12 )! (− 12 )! This is a rather distressing development. We were expecting an answer in simple form, but the answer that we are looking at contains some factorials of negative numbers, 46 The Hypergeometric Database and some of these negative numbers are negative integers, which are precisely the places where the factorial function is undefined. Fortunately, what we have is a ratio of two factorials at negative integers; if we take an appropriate limit, the singularities will cancel, and a pleasant limiting ratio will ensue.

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

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