Interesting Hypergeometric Theorems



                              By Shalosh B. Ekhad



           Let R(a,n) denote the rising factorial: a(a+1)...(a+n-1).

Theorem , 1, : ,

                                                         1/2 n
                                    1/2    (9/8 - 3/8 I 3   )  R(2/3, n)
    F(-n, n + 1, -2 n, 1/2 - 1/2 I 3   ) = -----------------------------
                                                     R(1/2, n)

                                                    /256\n
                                                    |---|  R(5/8, n) R(9/8, n)
                                                    \243/
 Theorem , 2, : , F(-n, -n + 1/4, 2 n + 5/4, 1/9) = --------------------------
                                                                    13
                                                        R(2/3, n) R(--, n)
                                                                    12

                                                    /256\n
                                                    |---|  R(7/8, n) R(11/8, n)
                                                    \243/
 Theorem , 3, : , F(-n, -n - 1/4, 2 n + 7/4, 1/9) = ---------------------------
                                                                    11
                                                        R(4/3, n) R(--, n)
                                                                    12

                                                         n
                                                    (1/4)  R(3/4, n) R(5/4, n)
  Theorem , 4, : , F(-2 n, n + 1/2, 2 n + 3/2, 2) = --------------------------
                                                       R(7/8, n) R(9/8, n)

Theorem , 5, : ,

                                   R(3/8, n) R(7/8, n) R(5/8, n) R(9/8, n)
    F(-2 n, 1/2, 4 n + 3/2, 1/4) = ---------------------------------------
                                                           11       13
                                   R(7/12, n) R(5/12, n) R(--, n) R(--, n)
                                                           12       12

Theorem , 6, : ,

                                            /    1/2     \n
                                            |54 5      27|
                                     1/2    |------- + --|  R(2/3, n) R(4/3, n)
                                    5       \  125     25/
    F(-2 n, 2 n + 1, 3 n + 2, 1/2 - ----) = -----------------------------------
                                     2              R(4/5, n) R(6/5, n)

                                                         /27\n
                                                         |--|  R(4/3, n)
                                                         \32/
       Theorem , 7, : , F(-n, -n + 1/3, n + 4/3, -1/8) = ---------------
                                                            R(7/6, n)

                                                     /64\n
                                                     |--|  R(3/4, n) R(5/4, n)
                                                     \81/
 Theorem , 8, : , F(-n, -n + 1/2, 2 n + 3/2, -1/3) = -------------------------
                                                        R(2/3, n) R(7/6, n)

                                                    /27\n
                                                    |--|  R(5/6, n) R(7/6, n)
                                                    \32/
  Theorem , 9, : , F(-n, n + 1/2, 3 n + 3/2, 1/2) = -------------------------
                                                       R(7/8, n) R(9/8, n)

                                                     /256\n
                                                     |---|  R(3/4, n) R(5/4, n)
                                                     \243/
 Theorem , 10, : , F(-n, -n + 1/2, 2 n + 3/2, 1/9) = --------------------------
                                                        R(5/6, n) R(7/6, n)

Theorem , 11, : ,

                                         /      1/2      \n
                                         |  50 5      125|
                                  1/2    |- ------- + ---|  R(3/5, n) R(2/5, n)
                                 5       \    27      27 /
    F(-2 n, 2 n + 1, -3 n, 1/2 - ----) = --------------------------------------
                                  2               R(1/3, n) R(2/3, n)

Theorem , 12, : ,

                                                   n
                                        (5 n + 2) 4  R(1/3, n) R(2/3, n)
    F(-3 n, -n + 3/2, -3 n + 1/2, -3) = --------------------------------
                                         (2 n - 1) R(1/6, n) R(5/6, n)

Theorem , 13, : ,

                                       /          1/2\n
                                       |125   25 5   |
                          1/2          |--- + -------|  R(3/5, n) R(2/5, n)
                         5             \128     128  /
    F(-2 n, n + 1, -4 n, ---- - 1/2) = ------------------------------------
                          2                    R(1/4, n) R(3/4, n)

Theorem , 14, : ,

                                          /      1/2     \n
                                          |  27 5      27|
                             1/2          |- ------- + --|  R(2/3, n) R(4/3, n)
                            5             \    250     50/
    F(-2 n, n + 1, 3 n + 2, ---- + 3/2) = -------------------------------------
                             2                     R(4/5, n) R(6/5, n)

Theorem , 15, : , F(-n, -n + 1/2, 4 n + 3/2, 1/5) =

    /16384\n
    |-----|  R(3/8, n) R(7/8, n) R(5/8, n) R(9/8, n)
    \15625/
    ------------------------------------------------
                                         11
        R(3/5, n) R(2/5, n) R(9/10, n) R(--, n)
                                         10

                                                     1/2
                                                    5
Theorem , 16, : , F(-2 n, n + 1/2, 3 n + 3/2, 3/2 - ----) =
                                                     2

    /         1/2\n
    |27   27 5   |
    |-- + -------|  R(5/6, n) R(7/6, n)
    \50     250  /
    -----------------------------------
                         11
            R(9/10, n) R(--, n)
                         10

                                                          /27\n
                                                          |--|  R(5/3, n)
                                                          \32/
       Theorem , 17, : , F(-n, -n - 1/3, n + 5/3, -1/8) = ---------------
                                                             R(3/2, n)

                                                         1/2
Theorem , 18, : , F(-2 n, 2 n + 1, 4 n + 2, 1/2 + 1/2 I 3   ) =

                    1/2 n
    (8/27 - 8/27 I 3   )  R(3/4, n) R(5/4, n)
    -----------------------------------------
               R(2/3, n) R(7/6, n)

Theorem , 19, : ,

                                                            1/2 n
                                       1/2    (2/3 + 2/9 I 3   )  R(3/2, n)
    F(-n, n + 1, 2 n + 2, 1/2 - 1/2 I 3   ) = -----------------------------
                                                        R(4/3, n)

                                     7.337