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).