A few identities (V)
with the Polygamma function
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

 

                             5
                      - 80 pi  - Psi(4, 1/4) + Psi(4, 3/4)

                            _______________________

                           5         5  1/2
    - 1488 Zeta(5) - 600 pi  - 352 pi  3    + 15 Psi(4, 3/4) - 2 Psi(4, 1/6)

                            _______________________

                       5  1/2
                  32 pi  3    + 3 Psi(4, 1/3) - 3 Psi(4, 2/3)

                            _______________________

                   32 Psi(4, 1/2) - Psi(4, 1/4) - Psi(4, 3/4)

                            _______________________

                       5  1/2
                  32 pi  3    + 3 Psi(4, 1/3) - 3 Psi(4, 2/3)

                            _______________________

         - 31 Psi(4, 1/3) - 31 Psi(4, 2/3) + Psi(4, 1/6) + Psi(4, 5/6)

                            _______________________

                             5  1/2
           96 Zeta(5) + 48 pi  3    + 9 Psi(4, 1/3) - 35 Psi(4, 1/2)

                            _______________________

                   32 Psi(4, 1/3) - Psi(4, 2/3) - Psi(4, 1/6)

                            _______________________

                        5  1/2
                 - 32 pi  3    + 93 Psi(4, 1/3) - 3 Psi(4, 1/6)

                            _______________________

                        5  1/2
                 - 32 pi  3    - 3 Psi(4, 1/3) + 3 Psi(4, 2/3)

                            _______________________

                         5
      288 Zeta(5) - 40 pi  - 66 Psi(4, 2/3) + Psi(4, 3/4) + 2 Psi(4, 5/6)

                            _______________________

                           5
                    - 40 pi  + 16 Psi(4, 1/2) - Psi(4, 1/4)

                            _______________________

                        5  1/2
                 - 32 pi  3    - 93 Psi(4, 2/3) + 3 Psi(4, 5/6)

                            _______________________

                          5
  - 1920 Zeta(5) - 1880 pi  + 99 Psi(4, 2/3) - 47 Psi(4, 1/4) + 3 Psi(4, 1/6)

                            _______________________

              5  1/2
      - 512 pi  3    - 465 Psi(4, 2/3) + 121 Psi(4, 1/4) + 121 Psi(4, 3/4)

           - 17 Psi(4, 1/6)

                            _______________________

              5  1/2
       - 32 pi  3    - 93 Psi(4, 2/3) + 121 Psi(4, 1/4) + 121 Psi(4, 3/4)

            - 16 Psi(4, 1/6) - 13 Psi(4, 5/6)

                            _______________________

             - 1023 Psi(4, 2/3) + 7744 Psi(4, 1/2) - 31 Psi(4, 1/6)

                            _______________________

                  - 32 Psi(4, 1/2) + Psi(4, 1/4) + Psi(4, 3/4)

                            _______________________

                  - 32 Psi(4, 1/3) + Psi(4, 2/3) + Psi(4, 1/6)

                            _______________________

                   32 Psi(4, 1/3) - Psi(4, 2/3) - Psi(4, 1/6)

                            _______________________

                  - 32 Psi(4, 1/3) + Psi(4, 2/3) + Psi(4, 1/6)

                            _______________________

                        5  1/2
                 - 32 pi  3    - 3 Psi(4, 1/3) + 3 Psi(4, 2/3)

                            _______________________

                      5  1/2
  - 72 Zeta(5) + 16 pi  3    + 63 Psi(4, 2/3) - 4 Psi(4, 1/2) - 2 Psi(4, 5/6)

                            _______________________

                          5
                     40 pi  - 16 Psi(4, 1/2) + Psi(4, 1/4)

                            _______________________

                  - 32 Psi(4, 1/2) + Psi(4, 1/4) + Psi(4, 3/4)

                            _______________________

                       5  1/2
                  32 pi  3    - 93 Psi(4, 1/3) + 3 Psi(4, 1/6)

                            _______________________

                   Psi(4, 1/3) - 32 Psi(4, 2/3) + Psi(4, 5/6)

                            _______________________

                   32 Psi(4, 1/3) - Psi(4, 2/3) - Psi(4, 1/6)

                            _______________________

          5
   - 40 pi  - 61 Psi(4, 1/4) - 60 Psi(4, 3/4) + 8 Psi(4, 1/6) + 8 Psi(4, 5/6)

                            _______________________

                  - Psi(4, 1/3) + 32 Psi(4, 2/3) - Psi(4, 5/6)

                            _______________________

                  - 32 Psi(4, 1/3) + Psi(4, 2/3) + Psi(4, 1/6)

                            _______________________

                  - Psi(4, 1/3) + 32 Psi(4, 2/3) - Psi(4, 5/6)

                            _______________________

                          5        5  1/2
     - 48 Zeta(5) - 800 pi  - 96 pi  3    - 51 Psi(4, 1/3) + 20 Psi(4, 3/4)

          - Psi(4, 5/6)

                            _______________________

                         5  1/2
                 - 176 pi  3    - 121 Psi(4, 1/2) + Psi(4, 5/6)

                            _______________________

                          5
                     40 pi  + 16 Psi(4, 1/2) - Psi(4, 3/4)

                            _______________________

                       5  1/2
                  32 pi  3    + 3 Psi(4, 1/3) - 3 Psi(4, 2/3)

                            _______________________

                          5
                     40 pi  + 16 Psi(4, 1/2) - Psi(4, 3/4)

                            _______________________

                          5  1/2
                    352 pi  3    + Psi(4, 1/6) - Psi(4, 5/6)

                            _______________________

              1023 Psi(4, 2/3) - 242 Psi(4, 1/2) - 31 Psi(4, 5/6)

                            _______________________

                       5  1/2
   - 48 Zeta(5) - 48 pi  3    + 42 Psi(4, 2/3) - 43 Psi(4, 1/2) - Psi(4, 5/6)

                            _______________________

                           5
                    - 40 pi  + 16 Psi(4, 1/2) - Psi(4, 1/4)

                            _______________________

                       5  1/2
                  32 pi  3    + 93 Psi(4, 2/3) - 3 Psi(4, 5/6)

                            _______________________

                        5  1/2
                 - 32 pi  3    - 3 Psi(4, 1/3) + 3 Psi(4, 2/3)

                            _______________________

                 - 242 Psi(4, 1/2) + Psi(4, 1/6) + Psi(4, 5/6)

                            _______________________

                        5  1/2
                 - 32 pi  3    - 93 Psi(4, 2/3) + 3 Psi(4, 5/6)

                            _______________________

                       5  1/2
                  32 pi  3    + 93 Psi(4, 2/3) - 3 Psi(4, 5/6)

                            _______________________

                1023 Psi(4, 2/3) - Psi(4, 1/6) - 32 Psi(4, 5/6)

                            _______________________

                          5
                     40 pi  + 16 Psi(4, 1/2) - Psi(4, 3/4)

                            _______________________

                  - 32 Psi(4, 1/3) + Psi(4, 2/3) + Psi(4, 1/6)

bytes used=2068540, alloc=851812, time=1.88
                            _______________________

                        5  1/2
                 - 32 pi  3    - 3 Psi(4, 1/3) + 3 Psi(4, 2/3)

                            _______________________

                          5
                     40 pi  - 16 Psi(4, 1/2) + Psi(4, 1/4)

                            _______________________

                  - 23808 Zeta(5) - Psi(4, 1/4) - Psi(4, 3/4)

                            _______________________

      121 Psi(4, 1/4) + 121 Psi(4, 3/4) - 16 Psi(4, 1/6) - 16 Psi(4, 5/6)

                            _______________________

                  - 32 Psi(4, 1/2) + Psi(4, 1/4) + Psi(4, 3/4)

                            _______________________

                        5  1/2
                 - 32 pi  3    + 93 Psi(4, 1/3) - 3 Psi(4, 1/6)

                            _______________________

                             5
                      - 80 pi  - Psi(4, 1/4) + Psi(4, 3/4)

                            _______________________

                            5
                       80 pi  + Psi(4, 1/4) - Psi(4, 3/4)

                            _______________________

                          5         5  1/2
    - 600 Zeta(5) - 320 pi  + 176 pi  3    - 33 Psi(4, 2/3) + 8 Psi(4, 3/4)

                            _______________________

                        5  1/2
                  176 pi  3    - 121 Psi(4, 1/2) + Psi(4, 1/6)

                            _______________________

                   32 Psi(4, 1/2) - Psi(4, 1/4) - Psi(4, 3/4)

                            _______________________

         - 144 Zeta(5) + 33 Psi(4, 1/3) - 250 Psi(4, 1/2) + Psi(4, 5/6)

                            _______________________

                  - 32 Psi(4, 1/3) + Psi(4, 2/3) + Psi(4, 1/6)

                            _______________________

                       5  1/2
                  32 pi  3    + 3 Psi(4, 1/3) - 3 Psi(4, 2/3)

                            _______________________

               5  1/2
        - 32 pi  3    + 93 Psi(4, 1/3) - 726 Psi(4, 1/2) + 3 Psi(4, 5/6)

                            _______________________

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

                            _______________________

                           5
                    - 40 pi  - 16 Psi(4, 1/2) + Psi(4, 3/4)

                            _______________________

                   Psi(4, 1/3) - 32 Psi(4, 2/3) + Psi(4, 5/6)

                            _______________________

                   Psi(4, 1/3) - 32 Psi(4, 2/3) + Psi(4, 5/6)

                            _______________________

                5  1/2
           32 pi  3    + 2 Psi(4, 1/3) + 29 Psi(4, 2/3) - Psi(4, 5/6)

                            _______________________

                          5
                     40 pi  + 16 Psi(4, 1/2) - Psi(4, 3/4)

                            _______________________

                         5         5  1/2
     336 Zeta(5) - 760 pi  + 416 pi  3    + 78 Psi(4, 1/3) - 19 Psi(4, 1/4)

                            _______________________

                       5  1/2
                  32 pi  3    - 93 Psi(4, 1/3) + 3 Psi(4, 1/6)

                            _______________________

                            5
                       80 pi  + Psi(4, 1/4) - Psi(4, 3/4)

                            _______________________

                          5
  - 1920 Zeta(5) - 1880 pi  + 99 Psi(4, 2/3) - 47 Psi(4, 1/4) + 3 Psi(4, 1/6)

                            _______________________

                       5  1/2
                  32 pi  3    + 3 Psi(4, 1/3) - 3 Psi(4, 2/3)

                            _______________________

                            5
                       80 pi  + Psi(4, 1/4) - Psi(4, 3/4)

                            _______________________

                        5  1/2
                  176 pi  3    - 121 Psi(4, 1/2) + Psi(4, 1/6)

                            _______________________

                          5         5  1/2
    - 600 Zeta(5) - 320 pi  + 176 pi  3    - 33 Psi(4, 2/3) + 8 Psi(4, 3/4)

                            _______________________

                  - Psi(4, 1/3) + 32 Psi(4, 2/3) - Psi(4, 5/6)

                            _______________________

                          5
                     40 pi  - 16 Psi(4, 1/2) + Psi(4, 1/4)

                            _______________________

                   Psi(4, 1/3) - 32 Psi(4, 2/3) + Psi(4, 5/6)

                            _______________________

          24 Zeta(5) + 5 Psi(4, 1/3) + 5 Psi(4, 2/3) - 39 Psi(4, 1/2)

                            _______________________

                          5
                     40 pi  - 16 Psi(4, 1/2) + Psi(4, 1/4)

                            _______________________

                          5  1/2
                    352 pi  3    + Psi(4, 1/6) - Psi(4, 5/6)

                            _______________________

                        5  1/2
                 - 32 pi  3    - 3 Psi(4, 1/3) + 3 Psi(4, 2/3)

                            _______________________

                        5  1/2
                  176 pi  3    - 121 Psi(4, 1/2) + Psi(4, 1/6)

                            _______________________

                          - 744 Zeta(5) - Psi(4, 1/2)

                            _______________________

                         5  1/2
                 - 176 pi  3    - 121 Psi(4, 1/2) + Psi(4, 5/6)

                            _______________________

                          5
                     40 pi  + 16 Psi(4, 1/2) - Psi(4, 3/4)

                            _______________________

                             5
                      - 80 pi  - Psi(4, 1/4) + Psi(4, 3/4)

                            _______________________

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

                            _______________________

                              5  1/2
          - 96 Zeta(5) - 48 pi  3    - 9 Psi(4, 1/3) + 35 Psi(4, 1/2)

                            _______________________

                   Psi(4, 1/3) - 32 Psi(4, 2/3) + Psi(4, 5/6)

                            _______________________

                           5
                    - 40 pi  + 16 Psi(4, 1/2) - Psi(4, 1/4)

                            _______________________

                           5
      1488 Zeta(5) + 600 pi  - 15 Psi(4, 3/4) + Psi(4, 1/6) + Psi(4, 5/6)

                            _______________________

                  - Psi(4, 1/3) + 32 Psi(4, 2/3) - Psi(4, 5/6)

                            _______________________

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

                            _______________________

                   32 Psi(4, 1/2) - Psi(4, 1/4) - Psi(4, 3/4)

                            _______________________