Complicated partition function emerges from simple constraints

I’m working on a statistical model which involves many degrees of freedom $?=1…?$. Each degree of freedom is described by a gamma distribution with its own parameters, which we will assume to be all different $v_i sim Gamma(k_i,theta_i)$ with $?_i>0$, $?_i>0$

I want to calculate the joint pdf of all the variables, given their sum:

$$ P(vec{v}|V) = frac{1}{Z_V} Big [ prod_{i=1}^{S} p(v_i) Big ] delta Big ( sum_{i=1}^{S}v_i – V Big ).$$

Now, my degrees o freedom are gamma-distributed: to be rigorous, I have to use incomplete gamma pdfs, in order to make sense of the integral. In my setting $v_i << V$ allowing us, if needed, to send $V to infty$.

$$ Z_V = frac{1}{prod_{i=1}^{S} gamma(V,k_i) theta^{k_i}}int_{0}^{V} dv_1 dots int_{0}^{V} dv_s prod_{i=1}^{S} v^{k_i-1}_i e^{-frac{v_i}{theta_i}} delta Big ( sum_{i=1}^{S}v_i – V Big ) .$$

Using the Fourier representation of the delta function to split the integrals one gets :

$$ Z_V propto int_{0}^{V} dv_1 dots int_{0}^{V} dv_s prod_{i=1}^{S} v^{k_i-1}_i e^{-frac{v_i}{theta_i}} int_{-infty}^{infty} dt e^{ it( sum_{i=1}^{S}v_i – V} ).$$

Each term in the productory is the moment generating function of a Gamma, and I end up with this integral

$$ Z_V propto int_{-infty}^{infty} dt e^{ – it V} frac{1}{prod_{i=1}^{S} (1-ittheta_i)^{k_i}}. $$

I can not go further with the problem. I tried some tricks, like Feynman and Schwinger parameterization, but it did not helped. Alternatively, one may consider it as the Fourier transform of the product of the generating functions and try to apply the residues theorem, but the $k_i$ s are not integers and, at least to me, is not clear how to deal with "fractional" singularities. Another thing that one may try is to introduce the grand canonical formulation of the problem.

Anyone has an idea about how to tackle this integral?