Does there exist an explicit expression for <munderover> &#x2211;<!-- ∑ --> <mrow class=

Roland Manning

Roland Manning

Answered question

2022-06-16

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?

Answer & Explanation

lorienoldf7

lorienoldf7

Beginner2022-06-17Added 19 answers

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 "closed form".

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