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Tk+1 /tk is a constant function of the summation index k. , are independent of the summation index k. Therefore a general geometric series looks like cxk . , in which tk+1 P (k) = , tk Q(k) where P and Q are polynomials in k. 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.

You have now identified the input series. 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.

Maple is a product of Waterloo Maple Software, Inc. 4 Mathematica is a product of Wolfram Research, Inc. 5 Axiom is a product of NAG (Numerical Algorithms Group), Ltd. 1) k+i 2j = , k j 0≤j≤n mm−1 exp  tm m! 2m s (n + 1)n−1 =1+ n≥1  = (−1)m (3m)! 2) tn , n! 4) Beautiful identities have often stimulated mathematicians to find correspondingly beautiful proofs for them; proofs that have perhaps illuminated the combinatorial or other significance of the equality of the two members, or possibly just dazzled us with their unexpected compactness and elegance.