A=B (symbolic summation algorithms) by Wilf, Zeilberger.

By Wilf, Zeilberger.

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2. ). To identify this series, note that the smallest value of k for which the term tk is nonzero is the term with k = −1. Hence we begin by shifting the origin of the sum as follows: 1 1 = . k≥−1 (2k + 1)(2k + 3)! k≥0 (2k − 1)(2k + 1)! The ratio of two consecutive terms is (k − 12 ) tk+1 1 = . 1) Hence our given series is identified as 1 − 12 = − 1F2 1 (2k + 1)(2k + 3)! 2 k 3 2 ; 1 . 3. Suppose we define the symbol [x, d]n =   n−1 (x − jd), if n > 0; 1, if n = 0. j=0 Now consider the series k n [x, d]k [y, d]n−k .

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! (k + p)! k=0 a hypergeometric function? The ratio of consecutive terms is (−1)k+1 ( x2 )2k+2+p k! (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 .

So let’s see how it does with the same summation problem. SumF. 2). So we have successfully identified the sum as a hypergeometric series. The next question is, does Hyp know how to evaluate this sum in simple form? To ask Hyp to look up your sum in its sum list we use the command SListe. SListe It replies by giving the numbers of the formulas in its database that might be of assistance in evaluating our sum. In this case its reply is to tell us that one of its four items S3202, S3231, S3232, S3233 might be of use.

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