This is qEKHAD, one of the Maple packages accompanying the forthcoming book "A=B" (soon to be published by A.K.Peters, Ltd.) by Marko Petkovsek, Herb Wilf, and Doron Zeilberger. Also try out the other packages: EKHAD and multiEKHAD The latest versions are always available by anon. ftp to ftp.math.temple.edu in the directory: /pub/zeilberg/programs or, on the Web, at http://www.math.temple.edu/~zeilberg For general help, and a list of the available functions, type "ezra();". For specific help type "ezra(procedure_name);" Warning: q is a global variable qfac(k) means (1-q)*(1-q^2)*...*(1-q^k), and NOT (1)(1+q)*...*(1+q+...+q^(k-1)) Version of July 1995 L( b(n,k)c(n,k) ) = c(n,k)B(n,k)-c(n,k-1)B(n,k-1) + H(n,k); where H(n,k)=y_1(n)b(n+1,k)G(n,k)+ y_2(n)b(n+2,k)[G(n,k)+G(n+1,k)] -F(n,k-1)B(n,k-1) ( use: F(n.k)=c(n,k+1)-c(n,k), G(n,k)=c(n+1,k)-c(n,k) ) Thus to prove L( b(n,k)c(n,k) ) = A(n,k)-A(n,k-1), we denote the second term in A(n,k) by D(n,k) and show that H(n,k) = D(n,k) - D(n,k-1). Regrouping the terms in H(n,k), we get: H(n,k) = k (1/2 k (k - 1)) (10 - k + 5 n) (12 + k + 7 n) (-1) q qfac(n + 2) a2 (q + q (2 k + 5 + 3 n) (k + 11 + 6 n) (5 n + k + 6) + 2 q + 2 q + q (6 + 4 n + 2 k) (9 n + 15) (2 n + 3) (10 n + 16) + q + q + q + q (3 n + 6) (10 - k + 6 n) (3 n + 4) (2 n + 4) - 2 q - 2 q + 3 q - q (12 - k + 7 n) (11 + 6 n) (6 + 4 n) (8 n + 14) - q - q + q + 2 q (8 - k + 5 n) (2 k + 4 n + 5) (6 n + 10) (6 n + 9) - 3 q - q + q - q (8 + 6 n) (10 + 5 n) (5 n + 9) (7 + 5 n) - 2 q - 2 q - 3 q - q (6 - k + 4 n) (6 + 2 k + 5 n) (4 - k + 2 n) - 2 q - q + 2 q (2 k + 6 n + 9) (4 n + 8) (2 k + 7 n + 11) (2 k + 6 n + 10) + q - 3 q - q - q (5 + 3 n) (7 + 5 n + 2 k) (k + 3 n + 4) (k + 5 n + 8) + 3 q + q + 2 q - 3 q (2 k + 3 n + 4) (k + 7 + 4 n) (k + 6 n + 9) - 2 q - 2 q - 2 q (2 k + 5 n + 8) (k + 13 + 8 n) (8 - k + 4 n) + 2 q + q + 2 q (6 - k + 3 n) (7 n + 12) (7 n + 10) (4 n + 5 + k) + 3 q + q - q + q (9 n + 14) (8 n + 13) (8 n + 12) (7 n + 10 + 2 k) + q + q + q + q (7 n + 10 + k) (2 k + 9 + 5 n) - q - 2 q ) / (n + 2) (n + 2) / (qfac(k) qfac(n + 2 - k) (q - 1) (1 + q )) / may set the constant factor to 1 while D(n,k) = -(q)_n+1/(q^2)_n+1 k 7 n 4 k n 3 4 k n k 2 2 n k 2 5 n 3 (- 2 + 2 q q (q ) + q (q ) q - q q q - 2 (q ) q q + (q ) q (q ) k k 2 n 2 3 k 2 4 n 2 k 6 n 4 k n 3 + q - (q ) (q ) q + 2 (q ) q (q ) + q q (q ) - q q q 6 n 4 k 2 k n 5 8 k n 2 k n 6 9 k 2 n 3 4 + q (q ) (q ) + q (q ) q - 2 q q q + q (q ) q - (q ) (q ) q n 5 n 3 n 2 4 n 2 3 n 2 k n 4 5 k - 2 q q - q (q ) - (q ) q - (q ) q - q q - q q + 2 (q ) q q 7 n 5 k n 2 3 k 5 n 3 k n 2 k n 2 3 + q (q ) q + 2 (q ) q q + q (q ) q ) (q ) b0 (- q + q q ) q k (1/2 k (k - 1)) (-1) q qfac(n + 2) / k (n + 2) (n + 1) / (q qfac(k) qfac(n + 2 - k) (q - 1) (1 - q )) / may set the constant factor to 1 This took 20.683 Seconds of CPU time