Interesting Hypergeometric Theorems By Shalosh B. Ekhad Let R(a,n) denote the rising factorial: a(a+1)...(a+n-1). 1/2 Theorem , 1, : , F(-n, n + 1, -2 n, 1/2 - 1/2 I 3 ) = 1/2 n 1/2 (9/8 - 3/8 I 3 ) GAMMA(n + 2/3) Pi ---------------------------------------- GAMMA(2/3) GAMMA(n + 1/2) Theorem , 2, : , F(-n, -n + 1/4, 2 n + 5/4, 1/9) = /256\n Pi |---| GAMMA(9/8 + n) GAMMA(7/8) GAMMA(5/8 + n) csc(----) GAMMA(2/3) \243/ 12 2/3 -------------------------------------------------------------------- Pi 13 11 csc(----) GAMMA(5/8) GAMMA(-- + n) GAMMA(--) GAMMA(n + 2/3) 8 12 12 Theorem , 3, : , F(-n, -n - 1/4, 2 n + 7/4, 1/9) = /256\n 1/2 11 |---| GAMMA(11/8 + n) GAMMA(5/8) GAMMA(7/8 + n) 3 GAMMA(--) 16 \243/ 12 -- --------------------------------------------------------------- 27 3 Pi 11 csc(----) GAMMA(7/8) GAMMA(n + 4/3) GAMMA(2/3) GAMMA(-- + n) 8 12 Theorem , 4, : , F(-2 n, n + 1/2, 2 n + 3/2, 2) = n 1/2 Pi (1/4) GAMMA(5/4 + n) 2 GAMMA(3/4 + n) csc(----) 8 1/4 --------------------------------------------------- GAMMA(9/8 + n) GAMMA(7/8 + n) Theorem , 5, : , F(-2 n, 1/2, 4 n + 3/2, 1/4) = 2/3 GAMMA(5/8 + n) 5 Pi Pi / / GAMMA(7/8 + n) GAMMA(9/8 + n) GAMMA(3/8 + n) csc(----) csc(----) / | 12 12 / \ Pi 3 Pi 13 11 csc(----) csc(----) GAMMA(5/12 + n) GAMMA(-- + n) GAMMA(-- + n) 8 8 12 12 \ GAMMA(7/12 + n)| / 1/2 5 Theorem , 6, : , F(-2 n, 2 n + 1, 3 n + 2, 1/2 - ----) = 2 / 1/2 \n |54 5 27| 1/2 Pi |------- + --| GAMMA(n + 2/3) GAMMA(n + 4/3) 3 csc(----) \ 125 25/ 5 3/10 ------------------------------------------------------------ GAMMA(6/5 + n) GAMMA(n + 4/5) Theorem , 7, : , /27\n 1/2 |--| GAMMA(n + 4/3) 3 GAMMA(2/3) \32/ F(-n, -n + 1/3, n + 4/3, -1/8) = 1/2 ------------------------------------ GAMMA(n + 7/6) GAMMA(5/6) Theorem , 8, : , F(-n, -n + 1/2, 2 n + 3/2, -1/3) = /64\n 1/2 |--| GAMMA(5/4 + n) 2 GAMMA(3/4 + n) GAMMA(2/3) \81/ 2/3 --------------------------------------------------- GAMMA(n + 7/6) GAMMA(5/6) GAMMA(n + 2/3) Theorem , 9, : , F(-n, n + 1/2, 3 n + 3/2, 1/2) = /27\n Pi |--| GAMMA(n + 7/6) GAMMA(5/6 + n) csc(----) \32/ 8 3/8 --------------------------------------------- GAMMA(9/8 + n) GAMMA(7/8 + n) Theorem , 10, : , F(-n, -n + 1/2, 2 n + 3/2, 1/9) = /256\n 1/2 |---| GAMMA(5/4 + n) 2 GAMMA(3/4 + n) \243/ 2/3 ----------------------------------------- GAMMA(n + 7/6) GAMMA(5/6 + n) 1/2 5 Theorem , 11, : , F(-2 n, 2 n + 1, -3 n, 1/2 - ----) = 2 / 1/2\n |125 50 5 | 1/2 |--- - -------| GAMMA(n + 3/5) GAMMA(n + 2/5) 3 \27 27 / 2/3 --------------------------------------------------- 2 Pi csc(----) GAMMA(n + 2/3) GAMMA(n + 1/3) 5 Theorem , 12, : , F(-3 n, -n + 3/2, -3 n + 1/2, -3) = n 1/2 (5 n + 2) 4 GAMMA(n + 2/3) GAMMA(n + 1/3) 3 ----------------------------------------------- (2 n - 1) GAMMA(5/6 + n) GAMMA(1/6 + n) 1/2 5 Theorem , 13, : , F(-2 n, n + 1, -4 n, ---- - 1/2) = 2 / 1/2\n |125 25 5 | 1/2 |--- + -------| GAMMA(n + 3/5) GAMMA(n + 2/5) 2 \128 128 / --------------------------------------------------- 2 Pi csc(----) GAMMA(3/4 + n) GAMMA(1/4 + n) 5 1/2 5 Theorem , 14, : , F(-2 n, n + 1, 3 n + 2, ---- + 3/2) = 2 / 1/2\n |27 27 5 | 1/2 Pi |-- - -------| GAMMA(n + 2/3) GAMMA(n + 4/3) 3 csc(----) \50 250 / 5 3/10 ------------------------------------------------------------ GAMMA(6/5 + n) GAMMA(n + 4/5) /16384\n Theorem , 15, : , F(-n, -n + 1/2, 4 n + 3/2, 1/5) = 4/5 |-----| GAMMA(5/8 + n) \15625/ Pi 2 Pi / / GAMMA(7/8 + n) GAMMA(9/8 + n) GAMMA(3/8 + n) csc(----) csc(----) / | 10 5 / \ Pi 3 Pi 11 csc(----) csc(----) GAMMA(-- + n) GAMMA(n + 2/5) GAMMA(n + 3/5) 8 8 10 \ GAMMA(9/10 + n)| / 1/2 5 Theorem , 16, : , F(-2 n, n + 1/2, 3 n + 3/2, 3/2 - ----) = 2 / 1/2\n |27 27 5 | Pi |-- + -------| GAMMA(n + 7/6) GAMMA(5/6 + n) csc(----) \50 250 / 10 3/10 ------------------------------------------------------- 11 GAMMA(-- + n) GAMMA(9/10 + n) 10 Theorem , 17, : , /27\n 1/2 |--| GAMMA(n + 5/3) Pi \32/ F(-n, -n - 1/3, n + 5/3, -1/8) = 3/4 -------------------------- GAMMA(2/3) GAMMA(n + 3/2) 1/2 Theorem , 18, : , F(-2 n, 2 n + 1, 4 n + 2, 1/2 + 1/2 I 3 ) = 1/2 n 1/2 (8/27 - 8/27 I 3 ) GAMMA(5/4 + n) 2 GAMMA(3/4 + n) GAMMA(2/3) 2/3 ------------------------------------------------------------------- GAMMA(n + 7/6) GAMMA(5/6) GAMMA(n + 2/3) 1/2 Theorem , 19, : , F(-n, n + 1, 2 n + 2, 1/2 - 1/2 I 3 ) = 1/2 n 1/2 1/2 (2/9 I 3 + 2/3) GAMMA(n + 3/2) Pi 3 4/9 --------------------------------------------- GAMMA(n + 4/3) GAMMA(2/3) 7.603