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))          (7 + 4 n + 2 k)      (6 + 4 n + 2 k)
- (-1)  q                a2 (- 2 q                + 3 q

          (5 + 3 n + 2 k)    (8 - k + 4 n)    (5 n + 8 + 2 k)      (8 + 4 n)
     + 2 q                + q              - q                - 2 q

          (5 - k + 3 n)    (5 + 4 n + 2 k)      (3 n + 4 + 2 k)
     + 2 q              - q                - 2 q

        (k + 5 + 4 n)    (2 n + 3)      (5 + 3 n)      (2 n + 4)      (6 + 4 n)
     + q              + q          - 3 q          + 3 q          - 7 q

        (7 + 4 n)      (5 n + 8)      (k + 6 + 4 n)    (k + 5 n + 8)
     - q          + 5 q          - 2 q              - q

        (5 n + 7 + k)      (3 n + 4 + k)      (6 n + 9)    (3 n + 6)
     - q              + 2 q              + 3 q          - q

          (3 n + 4)      (6 n + 10)    (7 - k + 4 n)    (7 + 5 n + 2 k)
     + 2 q          + 2 q           - q              + q

          (9 + k + 5 n)      (7 + 5 n)    (6 - k + 3 n)    (8 n + 13)
     + 2 q              - 2 q          + q              - q

          (7 n + 12)    (10 + k + 6 n)    (9 - k + 5 n)      (9 + 5 n)
     - 2 q           + q               - q              + 3 q

          (4 - k + 2 n)      (k + 7 + 4 n)
     - 2 q              - 2 q             )

       /                            (n + 1)        (n + 2)
      /  (qfac(k) qfac(n + 2 - k) (q        - 1) (q        - 1))
     /

                        may set the constant factor to 1



 while D(n,k) =

        n 2  4    n  2        k  8   n 5      7  k   n 4    k  6   n 4
     ((q )  q  + q  q  - 2 + q  q  (q )  + 2 q  q  (q )  + q  q  (q )

                n 2  4  k       n 3  4  k       n 2  3  k    n  3  k    k  n  2
          - 3 (q )  q  q  - 2 (q )  q  q  - 5 (q )  q  q  + q  q  q  - q  q  q

               k  n      k        k     n 3  5   k 2       n 2  4   k 2
          + 2 q  q  q + q  q + 3 q  - (q )  q  (q )  - 2 (q )  q  (q )

             3   n 2   k 2       k 2  2  n    n 2        k    n  2   3     k
          + q  (q )  (q )  + 2 (q )  q  q ) (q )  b0 (- q  + q  q ) q  (-1)

          (1/2 k (k - 1))   /   k
         q                 /  (q  qfac(k) qfac(n + 2 - k))
                          /

                        may set the constant factor to 1

                                   This took

                                     11.966

                              Seconds of CPU time