I am evaluating the partition function of a system of particles and incurred in sums like S ( a ) = ∑ k = 0 ∞ ( 2 k + 1 ) κ 2 e − ( 2 k + 1 ) a

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I am evaluating the partition function of a system of particles and incurred in sums like  S  (  a  )  =      ∑          k      =      0        ∞    (  2  k  +  1      )          κ      2            e          −      (      2      k      +      1      )      a

gorilomgl · Accepted Answer

I put a similar sum in Wolfram Alpha (Mathematica) and found:      ∑          k      =      0        m    (  2  k  +  1      )          κ              /            2            e          −      (      2      k      +      1      )      a        =      2          κ              /            2            e          −      3      a            (          e              2        a              Φ    (          e              −        2        a              ,    −          κ              /            2        ,    1          /        2    )    −          e              −        2        a        m              Φ    (          e              −        2        a              ,    −          κ              /            2        ,    m    +    3          /        2    )    )      ,where   Φ is the Lerch transcendent. The sum is said to converge for       e          R      e      (      a      )      Taking the limit as   m  →  ∞      ∑          k      =      0        ∞    (  2  k  +  1      )          κ              /            2            e          −      (      2      k      +      1      )      a        =      2          κ              /            2            e          −      a        Φ  (      e          −      2      a        ,  −      κ          /        2    ,  1      /    2  )This seems to be reasonable in the large a limit since  Φ  (      e          −      2      a        ,  −  κ      /    2  ,  1      /    2  )  =      ∑          ℓ      =      0        ∞              e              −        2        a        ℓ                    (      1              /            2      +      ℓ              )                  −          κ                      /                    2                    and the overall result in that large a limit is just:      e          −      a      as might be expected by looking at the k=0 term in the sum.

I am evaluating the partition function of a system of particles and incurred in sums like S(a)=sum_(k=0)^infty (2k+1)^(kappa/2)e^(-(2k+1)a)

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