If you type : zeilWZPv(binomial(n,k)*binomial(n+k,k),WZlog2,k,n,N); you would get: Theorem: Let b(, n, k, ) be defined by binomial(n, k) binomial(n + k, k) and let c(, n, k, ) be the potential function of the WZ pair (F,G) where F and G are k k (-1) n! k! (-1) n! k! 2 ---------------------, --------------------- (n + 1) (n + 1) (n + k + 1)! 2 (n + k + 1)! 2 Let a(, n, ) be the sum of F(, n, k, )*c(, n, k, ) this sum is annihilated by the operator 2 1, times, -n - 1 + (6 n + 9) N + (-n - 2) N `` Proof by Shalosh B. Ekhad We first find the linear recurrence operator annihilating the sum of b(, n, k, ), which is 2 -n - 1 + (6 n + 9) N + (-n - 2) N For a proof do zeil(b,k,n) in EKHAD. The certificate turned out to be 2 (2 n + 3) k 2 ------------------------- (-n - 1 + k) (-n - 2 + k) 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 k (-7 n - 13 + 2 k) - ------------------------------ 2 2 k + 2 n k - 22 - 7 n - 25 n 2 2 (2 n + 3) k -n - 1 + (6 n + 9) N + (-n - 2) N , 2 -------------------------, 1, (-n - 1 + k) (-n - 2 + k) k (-7 n - 13 + 2 k) - ------------------------------ 2 2 k + 2 n k - 22 - 7 n - 25 n If you type : zeilWZPv(binomial(n,k)^2*binomial(n+k,k),WZzeta2,k,n,N); you would get: Theorem: Let b(, n, k, ) be defined by 2 binomial(n, k) binomial(n + k, k) and let c(, n, k, ) be the potential function of the WZ pair (F,G) where F and G are (n + k) 2 (n + k) 2 (-1) (k!) (n - k - 1)! (-1) (k!) (n - k)! ------------------------------, 2 -------------------------- (n + k + 1)! (n + k + 1)! (n + 1) Let a(, n, ) be the sum of F(, n, k, )*c(, n, k, ) this sum is annihilated by the operator 2 2 2 2 1, times, -(n + 1) + (-11 n - 33 n - 25) N + (n + 2) 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 2 -(n + 1) + (-11 n - 33 n - 25) N + (n + 2) N For a proof do zeil(b,k,n) in EKHAD. The certificate turned out to be 2 2 3 (n + 1) (-11 n - 37 n - 30 + 6 n k + 7 k + k ) k -------------------------------------------------- 2 2 (-n - 1 + k) (-n - 2 + k) 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 3 2 2 2 - (-n + k) k (-11 n - 51 n - 76 n - 36 + 31 n k + 102 n k + 82 k - 17 n k 2 3 / 4 4 3 2 3 3 2 - 25 k + k ) / (n k + k - 57 k n - 18 n k - 45 k + 170 k / 2 2 2 3 2 2 4 3 + 217 n k + 331 n k + 48 n k - 417 n k - 42 n k - 225 n k 2 4 3 5 - 296 n k - 52 k - 124 n - 88 + 22 n + 64 n + 115 n + 11 n ) If you type : zeilWZPv(binomial(n,k)^2*binomial(n+k,k)^3,WZzeta3,k,n,N); you would get: Theorem: Let b(, n, k, ) be defined by 2 2 binomial(n, k) binomial(n + k, k) and let c(, n, k, ) be the potential function of the WZ pair (F,G) where F and G are k 2 k 2 (-1) (k!) (n - k - 1)! (-1) (k!) (n - k)! 1/2 ------------------------, --------------------- (n + k + 1)! (k + 1) 2 (n + k + 1)! (n + 1) Let a(, n, ) be the sum of F(, n, k, )*c(, n, k, ) this sum is annihilated by the operator 3 2 3 2 1, times, (n + 1) - (17 n + 51 n + 39) (2 n + 3) N + (n + 2) N `` Proof by Shalosh B. Ekhad We first find the linear recurrence operator annihilating the sum of b(, n, k, ), which is 3 2 3 2 (n + 1) - (17 n + 51 n + 39) (2 n + 3) N + (n + 2) N For a proof do zeil(b,k,n) in EKHAD. The certificate turned out to be 2 2 4 (2 n + 3) (-4 n - 12 n - 8 - 3 k + 2 k ) k 4 -------------------------------------------- 2 2 (-n - 1 + k) (-n - 2 + k) 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 2 3 3 4 - (-32 + 74 k - 96 n + 159 n k - 104 n - 81 k - 48 n + 21 k - 8 n 2 2 3 2 3 2 - 31 n k + 13 k n - 100 n k + 24 n k + 109 n k) (k + 1) k (-n + k) / 2 2 3 3 / ((72 + 64 k + 168 n + 58 n k + 218 n - 12 k + 224 n - 115 k / 4 2 2 4 3 4 5 6 + 154 n + 399 n k + 42 n k - 246 k n + 33 k + 56 n + 8 n 2 2 3 2 4 3 3 3 2 + 202 n k - 179 n k + 13 n k - 324 n k - 44 n k - 191 n k 3 2 4 4 2 5 + 256 n k - 175 n k + 55 n k - 32 n k) (n + 2)) If you type : gu:=WZchu(n,k,0,0,0,n); you would get: k 2 k 2 (-1) (k!) (n - k)! (-1) (k!) (n - k)! - -----------------------------------, 2 ---------------------------------- n n (n + k + 1) (n + k)! (-n + k) (-1) (n + 1) (n + k + 1) (n + k)! (-1) To use this WZ-pair to get a summation-acceleration formula type: lu:=Acc(gu,k,n); (-n) 2 (-1) GAMMA(n) 2 --------, 3 -------------- 2 GAMMA(2 n + 1) (n + 1) (-n) (-1) This means that that the sum of, 2 --------, from n=1 to infinity 2 (n + 1) equals 2 GAMMA(n) the sum of, 3 --------------, from n=1 to infinity GAMMA(2 n + 1) To check this automatically (and rigorously) dervied acceleration- convergence formula, type: CheckAcc(lu,n,30); 1.645941116, 1.644934066, .001007050 which should be small If you type : gu:=WZchu(n,k,0,0,0,4*n); you would get: k 2 (-1) (k!) (4 n - k)! 2 3 3 3 3 - -----------------------------------, 1/2 (240 n k + 128 n k + 25 k (4 n + k + 1) (4 n + k)! (-4 n + k) 3 2 2 2 3 2 2 + 140 k n + 130 k + 1200 n k + 640 n k + 708 n k + 335 k 2 5 3 4 2 + 6320 n k + 2336 n k + 2048 n k + 8704 n k + 6400 n k + 5360 n 4 5 3 k 2 + 1768 n + 6400 n + 2048 n + 230 + 8192 n ) (-1) (k!) (4 n - k)!/( (n + 1) (4 n + 3) (2 n + 1) (4 n + 1) (4 n + k + 2) (4 n + k + 1) (4 n + k)! (4 n + k + 3) (4 n + k + 4)) To use this WZ-pair to get a summation-acceleration formula type: lu:=Acc(gu,k,n); 2 4 5 3 115 + 2680 n + 884 n + 3200 n + 1024 n + 4096 n (n + 1) 1/8 ---------------------------------------------------, 1/10 (-1) 2 2 2 2 (n + 1) (4 n + 3) (2 n + 1) (4 n + 1) 5 4 3 2 2 (12608 n - 22960 n + 16384 n - 5781 n + 1001 n - 66) GAMMA(n) GAMMA(3 n - 2)/((4 n - 1) (2 n - 1) (4 n - 3) GAMMA(5 n)) This means that that the sum of, 2 4 5 3 115 + 2680 n + 884 n + 3200 n + 1024 n + 4096 n 1/8 ---------------------------------------------------, 2 2 2 2 (n + 1) (4 n + 3) (2 n + 1) (4 n + 1) from n=1 to infinity equals (n + 1) the sum of, 1/10 (-1) 5 4 3 2 2 (12608 n - 22960 n + 16384 n - 5781 n + 1001 n - 66) GAMMA(n) GAMMA(3 n - 2)/((4 n - 1) (2 n - 1) (4 n - 3) GAMMA(5 n)), from n=1 to infinity To check this automatically (and rigorously) dervied acceleration- convergence formula, type: CheckAcc(lu,n,30); 1.644869557, 1.644934066, -.000064509 which should be small