k (-1) n! k! For the kernel, ------------------------------, k (n + 1) (n + k + 1)! x (1 + x) a base for compatible WZ pairs with degree 1 is k k (1 + x) (-1) n! k! (-1) n! k! {[----------------------, --------------------], k (n + 1) k (n + 1) x %1 x (1 + x) %1 x (1 + x) k k (1 + x) (k x + n x + k + 1) (-1) n! k! (k x + n x + k) (-1) n! k! [---------------------------------------, ---------------------------]} 2 k (n + 1) k (n + 1) x %1 x (1 + x) x %1 x (1 + x) %1 := (n + k + 1)! and in Maple format {[(1+x)*(-1)^k*n!*k!/x/(n+k+1)!/(x^k)/((1+x)^(n+1)), (-1)^k*n!*k!/(n+k+1)!/(x^k )/((1+x)^(n+1))], [(1+x)*(k*x+n*x+k+1)/x^2*(-1)^k*n!*k!/(n+k+1)!/(x^k)/((1+x)^( n+1)), (k*x+n*x+k)/x*(-1)^k*n!*k!/(n+k+1)!/(x^k)/((1+x)^(n+1))]} ------------------------------------------ (n + k) 2 (-1) (k!) (n - k - 1)! For the kernel, ------------------------------, (n + k + 1)! (n + 1) a base for compatible WZ pairs with degree 1 is (n + k) 2 (n + k) 2 (-1) (k!) (n - k - 1)! (-2 k + 2 n) (-1) (k!) (n - k - 1)! {[------------------------------, -------------------------------------------]} (n + k + 1)! (n + k + 1)! (n + 1) and in Maple format {[(-1)^(n+k)*k!^2*(n-k-1)!/(n+k+1)!, (-2*k+2*n)*(-1)^(n+k)*k!^2*(n-k-1)!/(n+k+1 )!/(n+1)]} ------------------------------------------ k 2 (-1) (k!) (n - k - 1)! For the kernel, -----------------------------, 2 (n + k + 1)! (n + 1) (1 + k) a base for compatible WZ pairs with degree 3 is 2 k k 2 (n - k - 1)! (k!) (-1) 2 (-n + k) (-1) (k!) (n - k - 1)! lu := {[------------------------, - -----------------------------------]} (1 + k) (n + k + 1)! 2 (n + k + 1)! (n + 1) 2 k k 2 (n - k - 1)! (k!) (-1) 2 (-n + k) (-1) (k!) (n - k - 1)! {[------------------------, - -----------------------------------]} (1 + k) (n + k + 1)! 2 (n + k + 1)! (n + 1) and in Maple format {[(n-k-1)!*k!^2*(-1)^k/(1+k)/(n+k+1)!, -2*(-n+k)*(-1)^k*k!^2*(n-k-1)!/(n+k+1)!/ (n+1)^2]} ------------------------------------------