# numerical integration – infinite + integral sum

I have the following sum + integral to solve:
$$int_0 ^ { infty} dt , t ^ { frac {m-3} {2}} e ^ {- t (a ^ 2 + b ^ 2)} sum_ {n = – infty} ^ { infty} e ^ {- t (c ^ 2 n ^ 2 + 2bcn)}$$

My essay:

``````Refine[
Integrate[t^{(m - 3)/2} Exp[-t (a^2 + b^2)] Sum[
Exp[-t (c^2 n^2 + 2 b c n)], {n, -Infinity, Infinity}], {t, 0, Infinity}],
{Element[{a, b, c}, Reals]Element[m, Integers], m> 0}]
``````

or $$a, b, c in mathrm {R}$$, $$m in mathrm {N ^ + _ 0}$$.
Until now, this input gives an output with an integral (and EllipticTheta function):

$$text {Integrate} left[frac{sqrt{pi } t^{frac{m-3}{2}-frac{1}{2}} e^{b^2 t-t left(a^2+b^2right)} vartheta _3left(frac{b pi }{c},e^{-frac{pi ^2}{c^2 t}}right)}{left| cright| },{t,0,infty },text{Assumptions}to min mathbb{Z}land (a|b|c)in mathbb{R}land m>0right]$$

Is it possible to sum and integrate to receive a closed form, at least for some particular values ​​of $$m$$, as $$m = 0,1,2,3$$? How to change the entrance to achieve it?

I will accept any comments on how to improve this issue (and its title).

Posted on