By Wilf, Zeilberger.

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It was not until 1974, however, that the recognition mentioned above occurred. There was, therefore, a considerable time lag between the development of the “new technology” of hypergeometric identities, and its “application” to binomial coefficient sums of combinatorics. A similar, but much shorter, time lag took place before the third phase of the proof theory flowered. In the 1940s, the main ideas for the automated discovery of recurrence relations for hypergeometric sums were discovered by Sister Mary Celine Fasenmyer (see Chapter 4).

To do the routine verification, you now need only ask for In[6]:= FactorialSimplify[f[n+1,k]-f[n,k]-g[n,k+1]+g[n,k]], and after a few moments of reflection, you will be rewarded with Out[6]= 0 which is the name of the game. 5 29 A Maple session Now we’re going to try the same thing in Maple. ;, and the system responds by giving us back our input unaltered. So it needs to be coaxed. ); and we’re rewarded with the n + 1 that we were looking for. So Maple’s expand(); command is the way to simplify factorial expressions (in some versions of Maple this command does not work properly on quotients of products of factorials in which the factors are raised to powers).

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.