I'm looking for a closed-form for the following sum: ∑ n = 1 ∞ ( m n arccoth ( m n ) − 1 ) for | m | &gt; 1.

Question

I&#039;m looking for a closed-form for the following sum:      ∑          n      =      1              ∞            (    m    n        arccoth        (    m    n    )    −    1    )  for       |    m      |    &amp;gt;  1.

livin4him777lf · Accepted Answer

To get your result in terms of the Clausen function of order 2, we can use the identity                              ζ          ′                (        −        1        ,        x        )                            =        −        log        ⁡        G        (        x        +        1        )        +        x        log        ⁡        Γ        (        x        )        +                  ζ          ′                (        −        1        )                                          =        −        log        ⁡        Γ        (        x        )        −        log        ⁡        G        (        x        )        +        x        log        ⁡        Γ        (        x        )        +                  ζ          ′                (        −        1        )                                          =        −        log        ⁡        G        (        x        )        +        (        x        −        1        )        log        ⁡        Γ        (        x        )        +                  ζ          ′                (        −        1        )        ,            which holds for   x  &amp;gt;  0and the Barnes G reflection formula  log  ⁡      (                  G        (        1        −        x        )                    G        (        1        +        x        )              )    =  x  log  ⁡      (                  sin        ⁡        π        x            π        )    +                    Cl                  2                    ⁡      (      2      π      x      )              2      π        ,    0  &amp;lt;  x  &amp;lt;  1  ,which was used in skbmoore&#039;s answer to your previous question.Then                                          ζ            ′                                (            −            1            ,            1            −                          1              m                        )                    −                      ζ            ′                                (            −            1            ,            1            +                          1              m                        )                                              =                  −          log          ⁡                      (                                          G                                  (                  1                  −                                      1                    m                                    )                                                            G                                  (                  1                  +                                      1                    m                                    )                                                      )                    −                      1            m                    log          ⁡                      (            Γ                          (              1              +                              1                m                            )                        Γ                          (              1              −                              1                m                            )                        )                                                            =                  −          log          ⁡                      (                                          G                                  (                  1                  −                                      1                    m                                    )                                                            G                                  (                  1                  +                                      1                    m                                    )                                                      )                    −                      1            m                    log          ⁡                      (                          1              m                        Γ                          (                              1                m                            )                        Γ                          (              1              −                              1                m                            )                        )                                                            =                  −                      1            m                    log          ⁡                      (                                          sin                ⁡                                  (                                      π                    m                                    )                                            π                        )                    −                                                    Cl                                  2                                            ⁡                              (                                                      2                    π                                    m                                )                                                    2              π                                −                                    log              ⁡                              (                                  π                  m                                csc                ⁡                                  (                                      π                    m                                    )                                )                                      m                    .

I'm looking for a closed-form for the following sum: <munderover> ∑ <mr

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