By B. Jonsson
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Ak k We see that this cannot be the term ratio of a hypergeometric representation since the denominator has a zero root. This corresponds to the fact that a0 = 0, and any hypergeometric representation has a0 ∞= 0. By a suitable shift, however, we can overcome this difficulty and, as an important observation, the term ratio given shows us which shift will be successful! Since the denominator root is zero, we shift by one to eliminate it and to construct a (k + 1)-term. For bk : = ak+1 , we get ak+2 k+1−n bk+1 = =− bk ak+1 k+1 so that, from b0 = a1 = n, it follows that n k k=0 n k = n · 1 F0 −n + 1 −1 .
What is the order of this recurrence equation? 2 to determine ak+1 /ak for ak : = b(n + j, k)−b(n, k) for j = 1, . . , 3 when n , k n−k (b) b(n, k) = , k n , (c) b(n, k) = n k (d) b(n, k) = (n − k)! 11, Maple’s result n (−1)k k=0 n k √ n π2 2 = Γ 1+ 1 n 2 Γ 1 1 − n 2 2 was obtained. 4). 5). Which hypergeometric terms are determined by the right-hand sides of these identities? 8 to find hypergeometric representaB ak with largest possible summation range, if tions for k=A (a) ak = k(k − 1)(k − 2) (b) ak = n−k , k (c) ak = 1 k(k−1)(k−2) (d) ak = n , k 2n , k 2k .
5 Which of the following binomial sums are special cases of results in our database? 46 3 Hypergeometric Database n (a) k k=0 n (b) k=0 n (c) k=0 n (d) n , k (−1)k n k b+k , n (−1)k n k 2k , n (−1)k n k 3k . n k=0 In the affirmative cases, give the hypergeometric term results. Hint: If necessary, distinguish between odd and even n. 6 Find hypergeometric terms for the following sums using the database. n (a) k=0 n (b) (−1)k n k 2n − k , m−k (−1)k n k k+s . 7 Prove the following identities between hypergeometric functions using the database.