This is a small program, zeilWZP, that generalizes the so-called Zeilberger algorithm (a.k.a. as creative telescoping) to summands that are Closes Form times Potentials of WZ pairs written by Doron Zeilberger, in response to a question of Ken Ono First Written: Jan. 22, 1998 You MUST have Maple and the package EKHAD at the same directory. This program, as well as the famous package EKHAD are freely available from http://www.math.temple.edu/~zeilberg Please report all bugs to zeilberg@math.temple.edu The syntax is similar to zeil in EKHAD (see ezra(zeil); n EKHAD except that the WZ pair (F,G) is also an input the syntax is" zeilWZP(SUMMAND,F,G,k,n,N); Version of Feb. 2, 1998 This version corrects a subtle bug discovered by Helmut Prodinger Previous versions benefited from comments by Paula Cohen, Lyle Ramshaw, and Bob Sulanke. This is EKHAD, One of the Maple packages accompanying the book "A=B" (published by A.K. Peters, Welesley, 1996) by Marko Petkovsek, Herb Wilf, and Doron Zeilberger. The most current version is available on WWW at: http://www.math.temple.edu/~zeilberg . Information about the book, and how to order it, can be found in http://www.central.cis.upenn.edu/~wilf/AeqB.html . Please report all bugs to: zeilberg@math.temple.edu . All bugs or other comments used will be acknowledged in future versions. For general help, and a list of the available functions, type "ezra();". For specific help type "ezra(procedure_name)" Theorem: Let b(, n, k, ) be defined by 2 k ((n + k)!) ----------------- 4 2 (k!) ((n - k)!) and let c(, n, k, ) be the potential function of the WZ pair (F,G) where F and G are 2 2 2 4 k n + 4 k n + k + k + n + n n + 1 - 1/2 --------------------------------, 2 ----------------------- (n + k + 1) (k + 1) (n - k) k (n - k + 1) (n + k + 1) Let a(, n, ) be the sum of b(, n, k, )*c(, n, k, ) this sum is annihilated by the operator 1, times, 2 2 (2 n + 5) (n + 2) (3 n + 15 n + 19) (n + 1) 5 4 3 2 - (2 n + 3) (105 n + 1101 n + 4562 n + 9325 n + 9399 n + 3744) N 5 4 3 2 2 + (2 n + 5) (105 n + 999 n + 3746 n + 6923 n + 6319 n + 2284) N 2 2 3 - (2 n + 3) (n + 2) (3 n + 9 n + 7) (n + 3) N Proof by Shalosh B. Ekhad We first find the linear recurrence operator annihilating the sum of b(, n, k, ), which is 2 2 (2 n + 5) (n + 2) (3 n + 15 n + 19) (n + 1) 5 4 3 2 - (2 n + 3) (105 n + 1101 n + 4562 n + 9325 n + 9399 n + 3744) N 5 4 3 2 2 + (2 n + 5) (105 n + 999 n + 3746 n + 6923 n + 6319 n + 2284) N 2 2 3 - (2 n + 3) (n + 2) (3 n + 9 n + 7) (n + 3) N For a proof do zeil(b,k,n) in EKHAD. The certificate turned out to be 2 7 6 5 2 5 5 4 8 (2 n + 5) (k - 1) (48 n + 648 n - 24 n k + 60 n k + 3688 n + 11468 n 4 4 2 3 2 3 3 2 2 + 570 n k - 228 n k - 860 n k + 2152 n k + 21034 n - 1610 k n 2 2 2 2 + 4036 k n + 22743 n - 1498 k n + 3764 k n + 13410 n - 555 k + 3321 3 / 2 2 2 + 1398 k) k / ((n - k + 1) (n - k + 3) (n - k + 2) (n + 2)) / Applying this operator to c(n,k)*b(n,k) and subtrating c(n,k) times the operator applied on b(n,k), as was done in Doron Zeilberger's article: Closed Form (pun intended!) Contemporary Mathematics 143, pp 579-607, Theorem 9 (p. 600 ff) with slight notational changes to accomodate EKHAD's current convention of using forward differnce operators. We get that the operator S(N,n) promised by Theorem 9, is 1 The certificate certifying S(N,n), i.e. the one defining D(n,k), was found by EKHAD to be 2 3 3 2 (n - k) (k + 1) k (k - 1) (346287 n k - 83781 n + 54129 k + 79352 k n 4 3 4 5 6 - 8618 k n + 62507 k n + 203301 n k + 75172 n k + 17092 n k 7 8 2 2 2 + 2184 n k + 120 n k + 362786 k n + 213673 k n - 51440 k - 181381 k n 2 3 2 2 3 2 5 2 - 150975 n + 20382 k - 271165 k n - 222792 n k - 31420 n k 5 3 4 4 6 2 7 2 - 37212 n - 152679 n - 3330 k - 94935 n - 4992 n k - 336 n k 4 2 4 5 3 5 5 2 5 4 2 - 108622 n k + 12 n k + 96 n k + 286 k n + 376 k n - 8908 k n 5 4 6 3 5 3 4 4 4 3 - 120 n k + 324 n k + 3960 n k - 1176 n k + 19986 n k 3 4 3 3 6 5 8 7 - 4588 n k + 53322 n k - 8988 n + 185 k - 72 n - 1224 n - 19926) / 3 9 3 2 4 / (7047282 n k + 149904 n + 951471 n k + 4171810 k n + 1999583 k n / 3 4 5 6 + 1561720 k n + 12171727 n k + 14116015 n k + 11586042 n k 7 8 2 2 + 6902704 n k + 3007987 n k + 2458260 k n + 390312 k n - 454896 k 2 2 3 2 2 3 2 - 3357054 k n + 989064 n + 273726 k - 11090217 k n - 21762111 n k 5 2 5 3 4 4 - 25820039 n k + 6212694 n + 2946420 n + 359889 k + 5243606 n 6 2 7 2 8 2 9 2 - 16926626 n k - 8051132 n k - 2760333 n k - 665780 n k 4 2 7 3 6 4 4 5 - 28326668 n k + 1391848 n k + 1758936 n k - 3723976 n k 6 5 7 5 5 5 3 5 - 711594 n k - 160458 n k - 2004713 n k - 4567471 n k 5 2 5 4 2 10 - 3573656 k n - 1623144 k n + 4928600 k n + 213016 n k 8 3 7 4 11 10 2 9 3 + 394666 n k + 505068 n k + 32032 n k - 107300 n k + 74222 n k 8 4 5 4 6 3 5 3 + 93234 n k + 4120553 n k + 3427636 n k + 6037151 n k 4 4 4 3 3 4 3 3 + 6583134 n k + 7641810 n k + 7089031 n k + 6852259 n k 12 2 11 3 10 4 9 4 6 - 456 n k + 420 n k + 468 n k + 9972 n k + 295536 k n 12 11 2 10 3 6 2 13 + 2904 n k - 10380 n k + 8316 n k + 652633 k n + 120 n k 4 7 4 6 7 3 6 3 7 2 - 25042 n k + 598425 n k - 20166 k n + 800874 k n + 8628 k n 6 7 5 6 8 6 7 7 6 - 2868 n k + 279414 n k + 876 n k - 264 n k + 5165624 n 7 8 5 9 5 7 6 6 5 + 11936 k - 20844 n k - 1188 n k + 12756 n k + 56954 k - 327415 k 7 8 8 4 8 6 8 6 6 + 25160 k n - 8809 k n - 3515 k - 520 n k + 24 n k + 79774 n k 5 7 12 13 5 8 3 8 8 2 - 12280 n k + 1200 n + 48 n + 84 n k - 3646 n k - 8448 k n 9 10 11 8 7 9 + 420646 n + 92604 n + 13612 n + 1343486 n + 3092260 n + 185 k 4 9 9 3 9 2 9 + 12 n k + 96 k n + 286 k n + 376 k n)