How do I evaluate the following combination of random variables? Is it martingale? I'm about to analyse the following expression
$$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$
where $Y_j$ for all $j\in \mathbb{N}$, are independent and $\Gamma(\beta,1)$-distributed random variables and $X_i$, for all $i \in \mathbb{N}$ independent identically BetaDistr.($\alpha,\beta$). I have presumtion that this expression is a martingale and tried to prove the martingale property. Let $\mathcal{F_n}=\{X_i| Y_i: i \le n\}$
$$\mathbb{E}[Z_{n+1}|\mathcal{F}_n]=Z_n\mathbb{E}\left[\frac{\frac{Y_{n+1}}{\prod_{i=1}^{n+1} X_i}}{\sum_{j=1}^{n+1} \frac{Y_j}{\prod_{i=1}^j X_i}} |\mathcal{F}_n \right]=Z_n \frac{1}{\prod_{i=1}^n X_i}\mathbb{E}\left[\frac{\frac{Y_{n+1}}{X_{n+1}}}{\sum_{j=1}^{n} \frac{Y_j}{\prod_{i=1}^j X_i}+\frac{Y_{n+1}}{\prod_{i=1}^{n+1} X_i} } |\mathcal{F}_n \right]=?$$
I could calculate only till this step. Does somebody see how to proceed further? Maybe somebody has already had experience with this combination of random variables and could give me a hint what are another options to evaluate this expression?
 A: I assume that the family $\left(X_i,Y_j,i,j\geqslant 1\right)$ is independent.
There exists a formula for the conditional expectation of a function of two independent vectors with respect to the first vector. Let $f\colon \mathbb R^{k}\times\mathbb R^\ell\to\mathbb R$ be a measurable function and $U$ and $V$ two independent random vectors with values in $\mathbb R^{k}$ and $\mathbb R^\ell$ respectively. Then
$$
\mathbb E\left[f\left(U,V\right)\mid U\right]=g\left(U\right),
$$
where the function $g\colon\mathbb R^k\to\mathbb R$ is defined by $g\left(u\right)=\mathbb E\left[f\left(u,V\right) \right]$.
So it seems that a first step in the question is to compute 
$$
\mathbb E\left[
\frac{\frac{Y}{X}}{a+b\frac{Y}{X}}
\right]
$$
for fixed real numbers $a$ and $b$, where $Y$ has $\Gamma(\beta,1)$-distribution and $X$ is independent of $Y$ and has a $\beta$-distribution.
A: With the hint from @Davide Giraudo I started to investigate the expectation $\mathbb{E}\left[\frac{\frac{Y}{X}}{a+b\frac{Y}{X}}\right]$, where $Y$ has $\Gamma(\beta,1)$-distribution and $X$ BetaDistr.($\alpha,\beta$).
Let define $Z=\frac{Y}{X}$. 
In the paper of S. Nadarajah and S. Kotz "On the product and Ration of Gamma and Beta Random Variables" I could find the pdf of $Z=\frac{Y}{X}$ and it's given in paper as
$$f(z)=\frac{\lambda^{\beta}B(\beta+a,b)}{\Gamma(\beta)B(a,b)}z^{\beta-1}F(\beta+a;\beta+a+b;-\lambda z) \text{ for } z>0,$$
where $F(a;b;x)=\sum_{k=0}^{\infty} \frac{(a)_k}{(b)_k}\frac{x^k}{k!}$ is the confluent hypergeometric function, $\Gamma(\beta)$ express Gamma function and $Beta(\alpha,\beta)$ Beta function.
Adjusted to our case: $\beta=\beta$, $\lambda=1$, $a=\alpha, b=\beta$ it results to
$$f(z)=\frac{B(\beta+\alpha,\beta)}{\Gamma(\beta)B(\alpha,\beta)}z^{\beta-1}F(\beta+\alpha;\beta+\alpha+\beta;-z), \text{ for } z>0.$$
Like I understand I have to calculate the following integral
$\mathbb{E}\left[\frac{Z}{a+bZ}\right]=\int_{-\infty}^{\infty} \left( \frac{z}{a+bz}\right)\frac{B(\beta+\alpha,\beta)}{\Gamma(\beta)B(\alpha,\beta)}z^{\beta-1}F(\beta+\alpha;\beta+\alpha+\beta;-z)dz$ 
I'm working on that. I hope I'm on the right way.
A: I got stuck also in calculating of this expectation. :( 
$\mathbb{E}\left[\frac{Z}{a+bZ}\right]=\int_{-\infty}^{\infty} \left( \frac{z}{a+bz}\right)\frac{B(\beta+\alpha,\beta)}{\Gamma(\beta)B(\alpha,\beta)}z^{\beta-1}F(\beta+\alpha;\beta+\alpha+\beta;-z)dz$ 
What I tried to do to get feeling what result could be - I chose $\alpha=3$ and $\beta=2$.
$\frac{B(5,2)}{\Gamma(2)B(3,5)}\int_{-\infty}^{\infty} \left( \frac{z}{a+bz}\right)z^{2-1}F(5;7;-z)dz$
So I could calculate the coefficient before integral with WolframAlpha and it's $\frac{B(5,2)}{\Gamma(2)B(3,5)}=\frac{2}{5}$.
Like I see there're various formulas for the confluent hypergeometric function. I use this one.
$F(a;b;x)=\sum_{k=0}^{\infty} \frac{(a)_k}{(b)_k} \frac{x^k}{k!}$. In my case $a=\beta+\alpha=5$ and $b=\beta+\alpha+\beta=7$ are fixed (and integer).
Now I have 
$\frac{2}{5}\int_{-\infty}^{\infty} \left( \frac{z^2}{a+bz}\right)F(5,7,-z)dz$ 
and with the help of WolframAlpha (https://www.wolframalpha.com/input/?i=2%2F5++int+(z%5E2%2F(a%2Bbz))+Hypergeometric1F1%5B5,+7,+-z%5Ddz) I calculate the value of this integral.
The output looks a bit complex. I can not interpret what is says to my problem.
Does somebody see what I maybe did wrong or how can I incorporate this to the problem?
