For general help, and a list of the available functions, type "ezra();". For specific help type "ezra(procedure_name)" . The theorems are: n - 1 ----- \ 1 Theorem 1:, ) --------------- = / 2 ----- binomial(n, k) k = 0 /n - 1 \ |----- 3 2 j | 2 | \ (3 j + 12 j + 18 j + 10) 4 (j + 3/2)!| (n + 1)! (n + 1) | ) ----------------------------------------| | / 2 3 | |----- (j + 1) (j + 2) (j + 1)! | \j = 0 / ------------------------------------------------------------------ n (n + 3/2)! 4 / k (k - 1)\ n - 1 |- ---------| ----- \ 2 / \ q qrf(q, q, k) qrf(q, q, n - k) 2 Theorem 2:, ) -------------------------------------------- = qrf(q , q, n) / qrf(q, q, n) ----- k = 0 /n - 1 / / i (i - 1)\ |----- | |- ---------| | \ 2 2 | (i + 1) (3 i + 3) \ 2 / | ) qrf(q , q , i) \q + q + q | / |----- \i = 0 / (i + 1) (i - 4)\ / (i + 1) (i - 2)\ |- ---------------| 2 |- ---------------| \ 2 / (1 + 3/2 i - 1/2 i ) \ 2 / + q - q - q \ \ | | (2 i + 2)| / (i + 1) 2 (i + 2) | / - 2 q / / ((q - 1) qrf(q , q, i) (q - 1))| / / | / | / 2 2 qrf(q , q , n) m - 1 ----- \ 1 Theorem 3:, ) --------------------------------- = 1/2 (a + m - n + 1)! / binomial(m, k) binomial(a, n - k) ----- k = 0 / | | (a + 4)! (m + 1) | | | \ m - 1 / /n + i + i n + 1 2 i + a - n + 3 \ \\ ----- |2 |--------------- + ------------------| (a + i + 3)! (a - n + 2)!|| \ | \binomial(a, n) binomial(a, n - i)/ || ) |------------------------------------------------------------------|| / \ (a + i - n + 1)! (a + 4)! (i + 1) (i + 2) (a + i - n + 2) /| ----- | i = 0 / /((a + m + 3)! (a - n + 2)!) Theorem 4:, n - 1 ----- k \ (-1) ) -------------------------------------------------------------------- / binomial(n + b, n + k) binomial(n + c, c + k) binomial(b + c, b + k) ----- k = 0 /n - 1 |----- / | \ | = (b + n + 1) (c + n + 1) (b + c + 2)! n! | ) | | / \ |----- \i = 0 i 2 (-1) (b c + 3 b i + 3 b + 3 i c + 5 i + 12 i + 3 c + 7) --------------------------------------------------------- binomial(i + b, 2 i) binomial(b + c, i + b) \ b i c + b i + i c + b c + b + c + i + 1 | - --------------------------------------------------------| binomial(i + b, i) binomial(i + c, c) binomial(b + c, b)/ (b + c + i + 2)! (c + 1) (b + 1)/((2 i + 2) (c + i + 2) (b + i + 2) i! \ | | (c + i + 1) (b + i + 1) (b + c + 2)!)|/((b + 1) (c + 1) (b + c + n + 2)!) | | / {0} {0} {0} {0} this took, 1396.020, seconds.