Sum of Gamma distributions weighted by different multipoles -- 2

In the context of harmonic spherical decomposition where ?ℓ is the variance of ??? for a given multipole ℓ, I introduce a new observable "O" which the ratio between 2 sums depdening on multipoles "\ell" and corresponding C_\ell/C'_\ell. I tried to express this new observable as a function of Gamma function. For the numerator, the formulation of expectation is pretty simple with a sum on \ell and with the quantity C_\ell (which depends on \ell). For the denominator, I have got to compute the expectation of the inverse on double sum "\ell" and m : this is a little bit tricky but there exists a property which involves the Lapalace generating function. Indeed, computing E[1/X] leads to the computation of generating function M_x(t) and all the moments are given by derivatives of M_x(t) taking t=0. What it is disturbing is the shape parameters which depends on \ell (actually, this is (2\ell+1)/2). I don't know how to make appear a single Gamma function in denominator : that would make the calculations simpler. But I wonder if it is possible. From another side, there exists also a paper from P.G.Moschopoulos which is mentioned in comments. This method is relatively complex, but as I said, not impossible. This would allow me, If I can't have a single Gamma denominator (not summing over \ell), to calculate the linear combination of Gamma functions with factors (2l +1) in front of each corresponding C_l and doing the sum for N multipoles l (I use the notation \ell = l). Finally, I wonder if I have the right put the sum over l of (2l+1) into the shape parameter alpha of Gamma with alpha is this sum ? But in a first time, we need to compute the expectation E[1/B] to get E[A/B] = E[A] E[1/B] ( A and 1/B are not correlated). Any help/track/suggestion/remark is welcome. Best regards