Does there exist an explicit expression for ∑ k = 0 ∞ ( − 1 ) k e − λ ( 2 k + 1 ) 2 ( 2 k + 1 ) 3 ,where λ is a positive scalar?

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Does there exist an explicit expression for      ∑          k      =      0              ∞                                    (          −          1          )                          k                            e                  −          λ                                    (              2              k              +              1              )                                      2                                                          (        2        k        +        1        )                    3                ,where   λ is a positive scalar?

lorienoldf7 · Accepted Answer

The Jacobi theta function      θ    1    (  z  ,      e          −      4      λ        )  =  2      ∑          k      =      0        ∞    (  −  1      )    k        e          −      λ      (      2      k      +      1              )        2              sin  ⁡  (  (  2  k  +  1  )  z  )To get a factor of   1      /    (  2  k  +  1      )    3  , we can do some integration:      ∫    0          π            (                  π        2            16        −                  z        2            8        )        θ    1    (  z  ,      e          −      4      λ        )     d  z  =      ∑          k      =      0        ∞              (      −      1              )        k                    (      2      k      +      1              )        3                  e          −      λ      (      2      k      +      1              )        2            since       ∫    0          π            (                  π        2            8        −                  z        2            4        )    sin  ⁡  (  (  2  k  +  1  )  z  )     d  z  =  1      /    (  2  k  +  1      )    3  I rather doubt that the integral can be done in &quot;closed form&quot;.

Does there exist an explicit expression for <munderover> ∑ <mrow class=

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