A few identities (II)
with gamma
by
Simon Plouffe
Summer 1993


All equations are equal to 0 and where found using pari-Gp with an implementation of the LLL algorithm and an Interface with MapleV.

Identities : General case, gamma, Catalan, Zeta(3) and the Polygamma function

 

23 Psi(1/2) - 6 Psi(1/3) - 8 Psi(1/8) - 8 Psi(5/8) + 4 Psi(1/12) + 3 Psi(5/12)

            11
     - Psi(----) + 7 gamma
            12

_________________________________________________________________________________________

 41 Psi(1/2) + 6 Psi(1/3) - 8 Psi(1/8) - 8 Psi(5/8) + Psi(5/12) - 4 Psi(7/12)

               11
      - 3 Psi(----) + 25 gamma
               12

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/3) - Psi(2/3) - Psi(1/6) + gamma

_________________________________________________________________________________________

                Psi(1/2) - Psi(1/3) + 2 Psi(2/3) - Psi(5/6) + gamma

_________________________________________________________________________________________

                                                                        11
   6 Psi(1/2) - 3 Psi(1/3) + 3 Psi(2/3) - 4 Psi(1/4) + Psi(1/12) - Psi(----)
                                                                        12

        + 2 gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/3) - Psi(2/3) - Psi(1/6) + gamma

_________________________________________________________________________________________

               - Psi(1/2) - 2 Psi(1/3) + Psi(2/3) + Psi(1/6) - gamma

_________________________________________________________________________________________

               - Psi(1/2) - 2 Psi(1/3) + Psi(2/3) + Psi(1/6) - gamma

_________________________________________________________________________________________

               - Psi(1/2) - 2 Psi(1/3) + Psi(2/3) + Psi(1/6) - gamma

_________________________________________________________________________________________

                Psi(1/2) - Psi(1/3) + 2 Psi(2/3) - Psi(5/6) + gamma

_________________________________________________________________________________________

                Psi(1/2) - Psi(1/3) + 2 Psi(2/3) - Psi(5/6) + gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/3) - Psi(2/3) - Psi(1/6) + gamma

_________________________________________________________________________________________

                Psi(1/2) - Psi(1/3) + 2 Psi(2/3) - Psi(5/6) + gamma

_________________________________________________________________________________________

     - 3 Psi(1/2) - 4 Psi(1/3) + 2 Psi(2/3) + Psi(1/12) + Psi(7/12) - 3 gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/3) - Psi(2/3) - Psi(1/6) + gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/3) - Psi(2/3) - Psi(1/6) + gamma

_________________________________________________________________________________________

                Psi(1/2) - Psi(1/3) + 2 Psi(2/3) - Psi(5/6) + gamma

_________________________________________________________________________________________

     - 3 Psi(1/2) - 4 Psi(1/3) + 2 Psi(2/3) + Psi(1/12) + Psi(7/12) - 3 gamma

_________________________________________________________________________________________

                                                               11
     - 3 Psi(1/2) + 2 Psi(1/3) - 4 Psi(2/3) + Psi(5/12) + Psi(----) - 3 gamma
                                                               12

_________________________________________________________________________________________

                                                               11
     - 3 Psi(1/2) + 2 Psi(1/3) - 4 Psi(2/3) + Psi(5/12) + Psi(----) - 3 gamma
                                                               12

_________________________________________________________________________________________

                                                                          11
15 Psi(1/2) - 6 Psi(1/3) - 16 Psi(1/4) + 4 Psi(1/12) + 3 Psi(5/12) - Psi(----)
                                                                          12

     - gamma

_________________________________________________________________________________________

     - 15 Psi(1/2) + 6 Psi(1/3) + 16 Psi(1/4) - 4 Psi(1/12) - 3 Psi(5/12)

                 11
          + Psi(----) + gamma
                 12

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

 3 Psi(1/2) - 6 Psi(1/3) - 8 Psi(1/4) + 3 Psi(1/12) + 2 Psi(5/12) + Psi(7/12)

      - 5 gamma

_________________________________________________________________________________________

             3 Psi(1/2) + 3 Psi(1/3) - 2 Psi(1/6) - Psi(5/6) + 3 gamma

_________________________________________________________________________________________

               Psi(1/2) + 2 Psi(1/6) - Psi(1/12) - Psi(7/12) + gamma

_________________________________________________________________________________________

               Psi(1/2) + 2 Psi(1/6) - Psi(1/12) - Psi(7/12) + gamma

_________________________________________________________________________________________

                                                              11
      7 Psi(1/2) + 6 Psi(1/3) - 4 Psi(1/6) - Psi(5/12) - Psi(----) + 7 gamma
                                                              12

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

             - 7 Psi(1/2) + 2 Psi(3/4) + Psi(1/8) + Psi(5/8) - 3 gamma

_________________________________________________________________________________________

              7 Psi(1/2) - 2 Psi(3/4) - Psi(1/8) - Psi(5/8) + 3 gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(1/3) + 2 Psi(1/6) + Psi(5/6) - 3 gamma

_________________________________________________________________________________________

 21 Psi(1/2) + 6 Psi(1/3) - 8 Psi(3/4) - 3 Psi(1/12) - 2 Psi(5/12) - Psi(7/12)

      + 13 gamma

_________________________________________________________________________________________

  - 10 Psi(1/2) - 3 Psi(1/3) + 4 Psi(3/4) + Psi(1/6) + Psi(1/12) + Psi(5/12)

       - 6 gamma

_________________________________________________________________________________________

       - 9 Psi(1/2) + 3 Psi(1/3) + 8 Psi(3/4) - 3 Psi(5/6) + 2 Psi(5/12)

            - 2 Psi(7/12) - gamma

_________________________________________________________________________________________

             3 Psi(1/2) + 3 Psi(1/3) - 2 Psi(1/6) - Psi(5/6) + 3 gamma

_________________________________________________________________________________________

                                                              11
      7 Psi(1/2) + 6 Psi(1/3) - 4 Psi(1/6) - Psi(5/12) - Psi(----) + 7 gamma
                                                              12

_________________________________________________________________________________________

               Psi(1/2) + 2 Psi(1/6) - Psi(1/12) - Psi(7/12) + gamma

_________________________________________________________________________________________

      - 4 Psi(1/2) - 3 Psi(1/3) + Psi(5/6) + Psi(1/12) + Psi(7/12) - 4 gamma

_________________________________________________________________________________________

        - 8 Psi(1/2) + Psi(1/8) + Psi(3/8) + Psi(5/8) + Psi(7/8) - 4 gamma

_________________________________________________________________________________________

             3 Psi(1/2) + 3 Psi(1/3) - 2 Psi(1/6) - Psi(5/6) + 3 gamma

_________________________________________________________________________________________

                                                               11
     - 7 Psi(1/2) - 6 Psi(1/3) + 4 Psi(1/6) + Psi(5/12) + Psi(----) - 7 gamma
                                                               12

_________________________________________________________________________________________

             3 Psi(1/2) + 3 Psi(1/3) - 2 Psi(1/6) - Psi(5/6) + 3 gamma

_________________________________________________________________________________________

                                                        11
               Psi(1/2) + 2 Psi(5/6) - Psi(5/12) - Psi(----) + gamma
                                                        12

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(1/3) + 2 Psi(1/6) + Psi(5/6) - 3 gamma

_________________________________________________________________________________________

      - 4 Psi(1/2) - 3 Psi(1/3) + Psi(5/6) + Psi(1/12) + Psi(7/12) - 4 gamma

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + 2 Psi(5/6) - 3 gamma

_________________________________________________________________________________________

                                                              11
      - 4 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + Psi(5/12) + Psi(----) - 4 gamma
                                                              12

_________________________________________________________________________________________

              - Psi(1/2) - 2 Psi(1/6) + Psi(1/12) + Psi(7/12) - gamma

_________________________________________________________________________________________

                                                                          11
15 Psi(1/2) + 3 Psi(2/3) - 8 Psi(1/4) - 3 Psi(1/6) + 2 Psi(1/12) - 2 Psi(----)
                                                                          12

     + 7 gamma

_________________________________________________________________________________________

                                                                      11
 3 Psi(1/2) - 3 Psi(2/3) - 4 Psi(1/4) + 3 Psi(5/6) + Psi(1/12) - Psi(----) - gamma
                                                                      12

_________________________________________________________________________________________

             - 7 Psi(1/2) + 2 Psi(1/4) + Psi(3/8) + Psi(7/8) - 3 gamma

_________________________________________________________________________________________

                                                                          11
- 3 Psi(1/2) + 6 Psi(2/3) + 8 Psi(1/4) - 2 Psi(1/12) - 3 Psi(5/12) - Psi(----)
                                                                          12

     + 5 gamma

_________________________________________________________________________________________

              7 Psi(1/2) - 2 Psi(1/4) - Psi(3/8) - Psi(7/8) + 3 gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                                                                      11
 3 Psi(1/2) - 3 Psi(2/3) - 4 Psi(1/4) + 3 Psi(5/6) + Psi(1/12) - Psi(----) - gamma
                                                                      12

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/4) - Psi(1/8) - Psi(5/8) + gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/4) - Psi(1/8) - Psi(5/8) + gamma

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + 2 Psi(5/6) - 3 gamma

_________________________________________________________________________________________

- 17 Psi(1/2) - 3 Psi(2/3) + 8 Psi(3/4) - Psi(1/6) + 2 Psi(1/12) + 2 Psi(5/12)

     - 9 gamma

_________________________________________________________________________________________

              - Psi(1/2) - 2 Psi(1/6) + Psi(1/12) + Psi(7/12) - gamma

_________________________________________________________________________________________

               Psi(1/2) + 2 Psi(1/6) - Psi(1/12) - Psi(7/12) + gamma

_________________________________________________________________________________________

             - 7 Psi(1/2) + 2 Psi(3/4) + Psi(1/8) + Psi(5/8) - 3 gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

             3 Psi(1/2) + 3 Psi(2/3) - Psi(1/6) - 2 Psi(5/6) + 3 gamma

_________________________________________________________________________________________

                                                                          11
 9 Psi(1/2) - 3 Psi(2/3) - 8 Psi(3/4) + 3 Psi(1/6) - 2 Psi(1/12) + 2 Psi(----)
                                                                          12

      + gamma

_________________________________________________________________________________________

                                                                         11
  - 9 Psi(1/2) - 3 Psi(2/3) + 4 Psi(3/4) + 3 Psi(5/6) + Psi(1/12) - Psi(----)
                                                                         12

       - 5 gamma

_________________________________________________________________________________________

             - 7 Psi(1/2) + 2 Psi(3/4) + Psi(1/8) + Psi(5/8) - 3 gamma

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + 2 Psi(5/6) - 3 gamma

_________________________________________________________________________________________

             3 Psi(1/2) + 3 Psi(2/3) - Psi(1/6) - 2 Psi(5/6) + 3 gamma

_________________________________________________________________________________________

                                                              11
      - 4 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + Psi(5/12) + Psi(----) - 4 gamma
                                                              12

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + 2 Psi(5/6) - 3 gamma

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + 2 Psi(5/6) - 3 gamma

_________________________________________________________________________________________

            - 3 Psi(1/2) - 3 Psi(2/3) + Psi(1/6) + 2 Psi(5/6) - 3 gamma

_________________________________________________________________________________________

               - Psi(1/2) - 2 Psi(1/4) + Psi(1/8) + Psi(5/8) - gamma

_________________________________________________________________________________________

   5 Psi(1/2) - 4 Psi(1/4) - Psi(1/6) - Psi(5/6) + Psi(1/12) + Psi(5/12) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                     3 Psi(1/2) - Psi(1/4) - Psi(3/4) + gamma

_________________________________________________________________________________________

                    - 3 Psi(1/2) + Psi(1/4) + Psi(3/4) - gamma

_________________________________________________________________________________________

                                                        11
               Psi(1/2) + 2 Psi(5/6) - Psi(5/12) - Psi(----) + gamma
                                                        12

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(1/4) - Psi(1/8) - Psi(5/8) + gamma

_________________________________________________________________________________________

                                                                    11
 - 7 Psi(1/2) + 4 Psi(1/4) - Psi(1/6) - Psi(5/6) + Psi(7/12) + Psi(----) - 3 gamma
                                                                    12

_________________________________________________________________________________________

             - 7 Psi(1/2) + 2 Psi(1/4) + Psi(3/8) + Psi(7/8) - 3 gamma

_________________________________________________________________________________________

              - Psi(1/2) - 2 Psi(1/6) + Psi(1/12) + Psi(7/12) - gamma

_________________________________________________________________________________________

                                                        11
               Psi(1/2) + 2 Psi(5/6) - Psi(5/12) - Psi(----) + gamma
                                                        12

_________________________________________________________________________________________

              7 Psi(1/2) - 2 Psi(3/4) - Psi(1/8) - Psi(5/8) + 3 gamma

_________________________________________________________________________________________

                                                                        11
  - 12 Psi(1/2) + 8 Psi(3/4) + Psi(1/12) + Psi(5/12) - Psi(7/12) - Psi(----)
                                                                        12

       - 4 gamma

_________________________________________________________________________________________

              7 Psi(1/2) - 2 Psi(3/4) - Psi(1/8) - Psi(5/8) + 3 gamma

_________________________________________________________________________________________

               Psi(1/2) + 2 Psi(1/6) - Psi(1/12) - Psi(7/12) + gamma

_________________________________________________________________________________________

                                                                    11
   5 Psi(1/2) - 4 Psi(3/4) - Psi(1/6) - Psi(5/6) + Psi(7/12) + Psi(----) + gamma
                                                                    12

_________________________________________________________________________________________

                                                                          11
 - 11 Psi(1/2) + 8 Psi(3/4) + 2 Psi(1/6) + Psi(5/12) - 2 Psi(7/12) - Psi(----)
                                                                          12

      - 3 gamma

_________________________________________________________________________________________

             - 7 Psi(1/2) + 2 Psi(3/4) + Psi(1/8) + Psi(5/8) - 3 gamma

_________________________________________________________________________________________

                Psi(1/2) + 2 Psi(3/4) - Psi(3/8) - Psi(7/8) + gamma

_________________________________________________________________________________________

               Psi(1/2) + 2 Psi(1/6) - Psi(1/12) - Psi(7/12) + gamma

_________________________________________________________________________________________

                                                        11
               Psi(1/2) + 2 Psi(5/6) - Psi(5/12) - Psi(----) + gamma
                                                        12

_________________________________________________________________________________________

               Psi(1/2) + 2 Psi(1/6) - Psi(1/12) - Psi(7/12) + gamma

_________________________________________________________________________________________

                                                         11
              - Psi(1/2) - 2 Psi(5/6) + Psi(5/12) + Psi(----) - gamma
                                                         12

_________________________________________________________________________________________

 - 17 Psi(1/6) + Psi(5/6) - 2 Psi(3/8) - 2 Psi(7/8) + 8 Psi(1/12) - Psi(5/12)

      + 9 Psi(7/12) - 4 gamma

_________________________________________________________________________________________

      51 Psi(1/3) - 41 Psi(1/6) + 14 Psi(3/8) + 14 Psi(7/8) + 7 Psi(1/12)

                                   11
           - 5 Psi(5/12) - 12 Psi(----) + 28 gamma
                                   12

_________________________________________________________________________________________

  11 Psi(1/3) - 4 Psi(2/3) - 5 Psi(1/6) + 2 Psi(3/8) + 2 Psi(7/8) - Psi(7/12)

              11
       - Psi(----) + 4 gamma
              12

_________________________________________________________________________________________

 - 9 Psi(1/3) + 15 Psi(2/3) + 4 Psi(1/4) - 6 Psi(5/6) - Psi(5/12) + Psi(7/12)

      + 4 gamma

_________________________________________________________________________________________

      - 16 Psi(1/4) + 7 Psi(1/8) - Psi(3/8) + 7 Psi(5/8) - Psi(7/8) - 4 gamma

_________________________________________________________________________________________

  3 Psi(1/3) - 3 Psi(2/3) - 16 Psi(1/4) + 6 Psi(1/8) + 6 Psi(5/8) + Psi(5/12)

       - Psi(7/12) - 4 gamma

_________________________________________________________________________________________

       6 Psi(1/3) - 3 Psi(2/3) + Psi(1/4) + Psi(3/4) - 3 Psi(1/6) + 2 gamma

_________________________________________________________________________________________

       3 Psi(1/3) - 6 Psi(2/3) - Psi(1/4) - Psi(3/4) + 3 Psi(5/6) - 2 gamma

_________________________________________________________________________________________

  - Psi(1/3) - Psi(2/3) - 3 Psi(1/4) + Psi(3/4) + Psi(1/12) + Psi(5/12) - 2 gamma

_________________________________________________________________________________________

            - Psi(1/4) - 7 Psi(3/4) + 3 Psi(3/8) + 3 Psi(7/8) - 2 gamma

_________________________________________________________________________________________

14 Psi(1/3) - 7 Psi(2/3) + 2 Psi(1/4) - 7 Psi(1/6) + Psi(3/8) + Psi(7/8) + 4 gamma

_________________________________________________________________________________________

     - Psi(1/3) + 2 Psi(2/3) - 2 Psi(1/4) - Psi(5/6) + Psi(1/8) + Psi(5/8)

_________________________________________________________________________________________

 - 15 Psi(1/3) + 9 Psi(2/3) - 4 Psi(3/4) + 6 Psi(1/6) - Psi(5/12) + Psi(7/12)

      - 4 gamma

_________________________________________________________________________________________

                                                                         11
  - 7 Psi(1/3) + 2 Psi(2/3) - 4 Psi(3/4) + 3 Psi(1/6) + Psi(7/12) + Psi(----)
                                                                         12

       - 4 gamma

_________________________________________________________________________________________

 - 10 Psi(1/3) + 17 Psi(2/3) + 4 Psi(3/4) - 9 Psi(5/6) + Psi(1/12) + Psi(5/12)

      + 4 gamma

_________________________________________________________________________________________

                                                                         11
 - 9 Psi(1/3) + 15 Psi(2/3) + 4 Psi(3/4) - 6 Psi(5/6) + Psi(1/12) - Psi(----)
                                                                         12

      + 4 gamma

_________________________________________________________________________________________

                                                                         11
  - 2 Psi(1/3) + 7 Psi(2/3) + 4 Psi(3/4) - 3 Psi(5/6) - Psi(7/12) - Psi(----)
                                                                         12

       + 4 gamma

_________________________________________________________________________________________

14 Psi(1/3) - 28 Psi(2/3) - 6 Psi(3/4) - 3 Psi(1/8) - 3 Psi(5/8) + 7 Psi(5/12)

              11
     + 7 Psi(----) - 12 gamma
              12

_________________________________________________________________________________________

                                                                          11
 7 Psi(1/3) - 11 Psi(2/3) - 4 Psi(3/4) - Psi(1/12) + 2 Psi(5/12) + 3 Psi(----)
                                                                          12

      - 4 gamma

_________________________________________________________________________________________

                                                                        11
  - Psi(1/3) + 5 Psi(2/3) + 4 Psi(3/4) - Psi(5/12) - Psi(7/12) - 2 Psi(----)
                                                                        12

       + 4 gamma

_________________________________________________________________________________________

                                                                         11
  - 9 Psi(1/3) + 3 Psi(2/3) - 4 Psi(3/4) + 6 Psi(1/6) - Psi(1/12) + Psi(----)
                                                                         12

       - 4 gamma

_________________________________________________________________________________________

      - 21 Psi(1/3) - 22 Psi(1/4) + 21 Psi(1/6) + 3 Psi(3/8) + 3 Psi(7/8)

           + 7 Psi(5/12) - 7 Psi(7/12) - 16 gamma

_________________________________________________________________________________________

    - 39 Psi(1/3) - 28 Psi(1/4) + 33 Psi(1/6) + 10 Psi(5/12) - 7 Psi(7/12)

                  11
         + 3 Psi(----) - 28 gamma
                  12

_________________________________________________________________________________________

      33 Psi(1/3) + 28 Psi(1/4) - 15 Psi(1/6) - 7 Psi(1/12) - 9 Psi(5/12)

                    11
           - 2 Psi(----) + 28 gamma
                    12

_________________________________________________________________________________________

     - 3 Psi(1/3) + 8 Psi(1/4) + 33 Psi(5/6) - 17 Psi(5/12) + 2 Psi(7/12)

                    11
          - 15 Psi(----) + 8 gamma
                    12

_________________________________________________________________________________________

             7 Psi(1/4) + Psi(3/4) - 3 Psi(1/8) - 3 Psi(5/8) + 2 gamma

_________________________________________________________________________________________

       - 3 Psi(1/3) - 5 Psi(1/4) + 3 Psi(3/4) + 3 Psi(5/6) + 2 Psi(1/12)

                     11
            - 2 Psi(----) - 2 gamma
                     12

_________________________________________________________________________________________

            - Psi(1/4) - 7 Psi(3/4) + 3 Psi(3/8) + 3 Psi(7/8) - 2 gamma

_________________________________________________________________________________________

      - 18 Psi(1/3) - 7 Psi(1/4) - 7 Psi(3/4) + 12 Psi(1/6) + 3 Psi(5/12)

                    11
           + 3 Psi(----) - 14 gamma
                    12

_________________________________________________________________________________________

        3 Psi(1/3) + Psi(1/4) + Psi(3/4) - 2 Psi(1/6) - Psi(5/6) + 2 gamma

_________________________________________________________________________________________

       - 3 Psi(1/3) - Psi(1/4) - Psi(3/4) + 2 Psi(1/6) + Psi(5/6) - 2 gamma

_________________________________________________________________________________________

      - 21 Psi(1/3) - 6 Psi(3/4) + 14 Psi(1/6) + 7 Psi(5/6) - 3 Psi(1/8)

           - 3 Psi(5/8) - 12 gamma

_________________________________________________________________________________________

      39 Psi(1/3) + 28 Psi(3/4) - 33 Psi(1/6) + 7 Psi(1/12) - 3 Psi(5/12)

                     11
           - 10 Psi(----) + 28 gamma
                     12

_________________________________________________________________________________________

        3 Psi(1/3) - 8 Psi(1/6) - Psi(5/6) + 3 Psi(1/12) + 3 Psi(7/12)

_________________________________________________________________________________________

  - 3 Psi(1/3) + 6 Psi(3/4) + 2 Psi(1/6) + Psi(5/6) - 3 Psi(3/8) - 3 Psi(7/8)

_________________________________________________________________________________________

  - 3 Psi(1/3) + 6 Psi(3/4) + 2 Psi(1/6) + Psi(5/6) - 3 Psi(3/8) - 3 Psi(7/8)

_________________________________________________________________________________________

       - 3 Psi(1/3) + 8 Psi(1/6) + Psi(5/6) - 3 Psi(1/12) - 3 Psi(7/12)

_________________________________________________________________________________________

                                                                         11
 6 Psi(1/3) - 18 Psi(1/6) + 7 Psi(1/12) - Psi(5/12) + 7 Psi(7/12) - Psi(----)
                                                                         12

_________________________________________________________________________________________

       - 3 Psi(1/3) + 8 Psi(1/6) + Psi(5/6) - 3 Psi(1/12) - 3 Psi(7/12)

_________________________________________________________________________________________

                                                                   11
       3 Psi(1/3) - 2 Psi(1/6) - 7 Psi(5/6) + 3 Psi(5/12) + 3 Psi(----)
                                                                   12

_________________________________________________________________________________________

                                                                         11
  3 Psi(2/3) - 9 Psi(1/6) + 4 Psi(1/12) - Psi(5/12) + 4 Psi(7/12) - Psi(----)
                                                                         12

_________________________________________________________________________________________

 - 3 Psi(2/3) + 22 Psi(1/4) + 3 Psi(5/6) - 9 Psi(1/8) - 9 Psi(5/8) - Psi(5/12)

      + Psi(7/12) + 4 gamma

_________________________________________________________________________________________

             7 Psi(1/4) + Psi(3/4) - 3 Psi(1/8) - 3 Psi(5/8) + 2 gamma

_________________________________________________________________________________________

             Psi(1/4) + 7 Psi(3/4) - 3 Psi(3/8) - 3 Psi(7/8) + 2 gamma

_________________________________________________________________________________________

 3 Psi(2/3) - Psi(1/4) + 3 Psi(3/4) - 3 Psi(5/6) + Psi(5/12) - Psi(7/12) + 2 gamma

_________________________________________________________________________________________

             7 Psi(1/4) + Psi(3/4) - 3 Psi(1/8) - 3 Psi(5/8) + 2 gamma

_________________________________________________________________________________________

                                                                          11
 6 Psi(2/3) - 5 Psi(1/4) + 11 Psi(3/4) + Psi(1/12) - 3 Psi(7/12) - 4 Psi(----)
                                                                          12

      + 6 gamma

_________________________________________________________________________________________

     - Psi(1/4) - Psi(3/4) - 6 Psi(1/6) + 3 Psi(1/12) + 3 Psi(7/12) - 2 gamma

_________________________________________________________________________________________

                                                                        11
   - 3 Psi(2/3) - 3 Psi(1/4) + Psi(3/4) + 3 Psi(5/6) + Psi(1/12) - Psi(----)
                                                                        12

        - 2 gamma

_________________________________________________________________________________________

       - 3 Psi(2/3) - Psi(1/4) - Psi(3/4) + Psi(1/6) + 2 Psi(5/6) - 2 gamma

_________________________________________________________________________________________

     - 33 Psi(2/3) - 28 Psi(1/4) + 15 Psi(5/6) + 9 Psi(1/12) + 7 Psi(5/12)

          + 2 Psi(7/12) - 28 gamma

_________________________________________________________________________________________

      - 16 Psi(1/4) + 7 Psi(1/8) - Psi(3/8) + 7 Psi(5/8) - Psi(7/8) - 4 gamma

_________________________________________________________________________________________

                                                                        11
   6 Psi(2/3) + 4 Psi(3/4) - 3 Psi(1/6) - 3 Psi(5/6) + Psi(1/12) - Psi(----)
                                                                        12

        + 4 gamma

_________________________________________________________________________________________

 6 Psi(2/3) + 4 Psi(3/4) - Psi(1/6) - 5 Psi(5/6) + Psi(5/12) - Psi(7/12) + 4 gamma

_________________________________________________________________________________________

      21 Psi(2/3) - 26 Psi(3/4) - 21 Psi(1/6) + 15 Psi(1/8) + 15 Psi(5/8)

           - 14 Psi(5/12) + 14 Psi(7/12) + 4 gamma

_________________________________________________________________________________________

      - 21 Psi(2/3) - 6 Psi(3/4) + 7 Psi(1/6) + 14 Psi(5/6) - 3 Psi(1/8)

           - 3 Psi(5/8) - 12 gamma

_________________________________________________________________________________________

    16 Psi(1/4) + Psi(1/6) - Psi(5/6) - 6 Psi(1/8) - 6 Psi(5/8) - Psi(1/12)

                11
         + Psi(----) + 4 gamma
                12

_________________________________________________________________________________________

 7 Psi(1/4) - 5 Psi(3/4) + 3 Psi(1/6) + 3 Psi(5/6) - 3 Psi(1/12) - 3 Psi(5/12)

      + 2 gamma

_________________________________________________________________________________________

             Psi(1/4) + 7 Psi(3/4) - 3 Psi(3/8) - 3 Psi(7/8) + 2 gamma

_________________________________________________________________________________________

             Psi(1/4) + 7 Psi(3/4) - 3 Psi(3/8) - 3 Psi(7/8) + 2 gamma

_________________________________________________________________________________________

                                                                          11
 5 Psi(1/4) - 7 Psi(3/4) - 3 Psi(1/6) - 3 Psi(5/6) + 3 Psi(7/12) + 3 Psi(----)
                                                                          12

      - 2 gamma

_________________________________________________________________________________________

      11 Psi(1/4) - 13 Psi(3/4) - 6 Psi(5/6) - 3 Psi(1/12) + 3 Psi(7/12)

                    11
           + 6 Psi(----) - 2 gamma
                    12

_________________________________________________________________________________________

            - 7 Psi(1/4) - Psi(3/4) + 3 Psi(1/8) + 3 Psi(5/8) - 2 gamma

_________________________________________________________________________________________

            - 7 Psi(1/4) - Psi(3/4) + 3 Psi(1/8) + 3 Psi(5/8) - 2 gamma

_________________________________________________________________________________________

             Psi(1/4) + 7 Psi(3/4) - 3 Psi(3/8) - 3 Psi(7/8) + 2 gamma

_________________________________________________________________________________________

            - 7 Psi(1/4) - Psi(3/4) + 3 Psi(1/8) + 3 Psi(5/8) - 2 gamma

_________________________________________________________________________________________

            - 7 Psi(1/4) - Psi(3/4) + 3 Psi(1/8) + 3 Psi(5/8) - 2 gamma

_________________________________________________________________________________________

             7 Psi(1/4) + Psi(3/4) - 3 Psi(1/8) - 3 Psi(5/8) + 2 gamma

_________________________________________________________________________________________

            - 7 Psi(1/4) - Psi(3/4) + 3 Psi(1/8) + 3 Psi(5/8) - 2 gamma

_________________________________________________________________________________________

             Psi(1/4) + 7 Psi(3/4) - 3 Psi(3/8) - 3 Psi(7/8) + 2 gamma

_________________________________________________________________________________________

            - Psi(1/4) - 7 Psi(3/4) + 3 Psi(3/8) + 3 Psi(7/8) - 2 gamma

_________________________________________________________________________________________

            - 7 Psi(1/4) - Psi(3/4) + 3 Psi(1/8) + 3 Psi(5/8) - 2 gamma

_________________________________________________________________________________________

             Psi(1/4) + 7 Psi(3/4) - 3 Psi(3/8) - 3 Psi(7/8) + 2 gamma

_________________________________________________________________________________________

                                                                        11
  2 Psi(3/4) + 14 Psi(5/6) + Psi(1/8) + Psi(5/8) - 7 Psi(5/12) - 7 Psi(----)
                                                                        12

       + 4 gamma

_________________________________________________________________________________________

                                                                          11
 - 4 Psi(3/4) + Psi(1/6) - 13 Psi(5/6) - Psi(1/12) + 6 Psi(5/12) + 7 Psi(----)
                                                                          12

      - 4 gamma

_________________________________________________________________________________________