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Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: When can the sum of (1/p)th of the binomial coefficients have closed form? | When can the sum of 1 p th of the binomial coefficients have closed form Marko Petkovsek Herbert S. Wilf University of Ljubljana University of Pennsylvania Ljubljana Slovenia Philadelphia PA 19104-6395 Submitted May 23 1996 Accepted November 25 1996 Abstract We find all nonnegative integer r s p for which the sum Vk rn k has closed form. Let f -X pn fp r n X k k 0 k . where 0 r p are fixed integers. This is a definite sum in the sense that the summation limits and the summand are not independent. As we all know fr r n 2r hr - n 1 n Thus fr r n is a hypergeometric term and f2r r n is a linear combination of two hypergeometric terms. Supported in part by the Office of Naval Research 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 4 no. 2 1997 R21 2 Following PWZ let us say that a function f n has closed form if there is a hxed integer m and hypergeometric terms ti n 1 such that f n ZX1 ti n for all sufficiently large n. Our main results are as follows. Theorem 1 Let 0 r p and p 2r. Then fpr n does not have closed form. Theorem 2 Let 0 r s p be fixed integers. Then Sptr s n sn X k rn pn k does not have closed form unless r 0 p 2s or p s 2r or r 0 p s. 1 Reduction to an indefinite sum We begin by briefly discussing the method. One might anticipate that we would first find a recurrence formula that say fpr n satisfies using Zeilberger s algorithm and then prove using Petkovsek s theorem that the recurrence has no closed form solution. As described in PWZ this method is quite effective in many cases. In the present situation however the recurrence that fPir n satisfies will grow in complexity with p r. So for each fixed p r the argument would work but without further human input it could not produce a general proof i.e. a proof for all p r. This is somewhat analogous to the sums of the pth powers of all of the binomial coefficients of order n. There too the methods described in PWZ can show that no closed form exists for specific fixed values of p but the general question .
