Double sum of Lambert series: Partial sum in closed form desired!












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$begingroup$


We desire the things stated in the title for:



$sum_{k=2}^m sum_{n=1}^{k-1} {q^nover {1-q^n}}$



Some things I've looked into that may be of some help:



The first sum is just a truncated (partially summed) Lambert series, so take a look at the truncated Lambert series.



Answer #2 from this other question, which may be able to help using theta functions



If you're going to jump into using Mathematica or Sage, check out the q-polygamma functions, specifically q-digamma



More on the q-digamma function, check out Wolfram Alpha's entry too






sequences-and-series theta-functions q-series q-analogs







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    We desire the things stated in the title for:



    $sum_{k=2}^m sum_{n=1}^{k-1} {q^nover {1-q^n}}$



    Some things I've looked into that may be of some help:



    The first sum is just a truncated (partially summed) Lambert series, so take a look at the truncated Lambert series.



    Answer #2 from this other question, which may be able to help using theta functions



    If you're going to jump into using Mathematica or Sage, check out the q-polygamma functions, specifically q-digamma



    More on the q-digamma function, check out Wolfram Alpha's entry too






    sequences-and-series theta-functions q-series q-analogs







    share|cite|improve this question











    $endgroup$















      0












      0








      0


      2



      $begingroup$


      We desire the things stated in the title for:



      $sum_{k=2}^m sum_{n=1}^{k-1} {q^nover {1-q^n}}$



      Some things I've looked into that may be of some help:



      The first sum is just a truncated (partially summed) Lambert series, so take a look at the truncated Lambert series.



      Answer #2 from this other question, which may be able to help using theta functions



      If you're going to jump into using Mathematica or Sage, check out the q-polygamma functions, specifically q-digamma



      More on the q-digamma function, check out Wolfram Alpha's entry too






      sequences-and-series theta-functions q-series q-analogs







      share|cite|improve this question











      $endgroup$




      We desire the things stated in the title for:



      $sum_{k=2}^m sum_{n=1}^{k-1} {q^nover {1-q^n}}$



      Some things I've looked into that may be of some help:



      The first sum is just a truncated (partially summed) Lambert series, so take a look at the truncated Lambert series.



      Answer #2 from this other question, which may be able to help using theta functions



      If you're going to jump into using Mathematica or Sage, check out the q-polygamma functions, specifically q-digamma



      More on the q-digamma function, check out Wolfram Alpha's entry too






      sequences-and-series theta-functions q-series q-analogs




      sequences-and-series theta-functions q-series q-analogs






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 21 at 5:43







      user3108815

















      asked Jan 21 at 5:25









      user3108815user3108815

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      468






















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