I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:

$$R\left(n,m,j\right)=\sum_{k=0}^{n}a^{k}b^{n-k}c^{j-k}d^{k+m-j}\binom{j}{k}\binom{n+m-j}{n-k}\binom{n}{k}\binom{m}{j-k}$$

for positive real numbers $a,b,c,d\leq1$ and positive integers $n$, $m$, and $j$ with $j\leq m+n$.

The probability that the collision involved $n$ particles is given by $\left(1-\delta\right)\left(\delta^{n}\right)$ and the probability that collision involved $m$ particles is given by $\left(1-\delta\right)\left(\delta^{m}\right)$, for a positive real number $\delta<1$. So I believe that the probability of forming $j$ particles in the collision is given by

$$Pr\left(j\right)=\left(1-\delta\right)^{2}\sum_{n=0}^{\infty}\delta^{n}\sum_{m=0}^{\infty}\delta^{m}R\left(n,m,j\right)$$

Question: Is there a simple form for $Pr\left(j\right)$? One that does not depend on $n$ or $m$?

I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:

$$R\left(n,m,j\right)=\sum_{k=0}^{n}a^{k}b^{n-k}c^{j-k}d^{k+m-j}\binom{j}{k}\binom{n+m-j}{n-k}\binom{n}{k}\binom{m}{j-k}$$

for positive real numbers $a,b,c,d\leq1$ and positive integers $n$, $m$, and $j$ with $j\leq m+n$.

The probability that the collision involved $n$ particles is given by $\left(1-\delta\right)\left(\delta^{n}\right)$ and the probability that collision involved $m$ particles is given by $\left(1-\delta\right)\left(\delta^{m}\right)$, for a positive real number $\delta<1$.

Then what is the probability of forming $j$ particles, $Pr\left(j\right)$? Is there a simple form for $Pr\left(j\right)$ that does not depend on $n$ or $m$?

I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:

$$R\left(n,m,j\right)=\sum_{k=0}^{n}a^{k}b^{n-k}c^{j-k}d^{k+m-j}\binom{j}{k}\binom{n+m-j}{n-k}\binom{n}{k}\binom{m}{j-k}$$

for positive real numbers $a,b,c,d\leq1$ and positive integers $n$, $m$, and $j$ with $j\leq m+n$.

The probability that the collision involved $n$ particles is given by $\left(1-\delta\right)\left(\delta^{n}\right)$ and the probability that collision involved $m$ particles is given by $\left(1-\delta\right)\left(\delta^{m}\right)$, for a positive real number $\delta<1$. So I believe that the probability of forming $j$ particles in the collision is given by

$$Pr\left(j\right)=\left(1-\delta\right)^{2}\sum_{n=0}^{\infty}\delta^{n}\sum_{m=0}^{\infty}\delta^{m}\sum_{j=0}^{n+m}R\left(n,m,j\right)$$

Question: Is there a simple form for $Pr\left(j\right)$? One that does not depend on $n$ or $m$?

I am working on a certain collision problem. The probability of forming $j$ particles upon collision of $m$ and $n$ particles is given by the following equation:

$$R\left(n,m,j\right)=\sum_{k=0}^{n}a^{k}b^{n-k}c^{j-k}d^{k+m-j}\binom{j}{k}\binom{n+m-j}{n-k}\binom{n}{k}\binom{m}{j-k}$$

for positive real numbers $a,b,c,d\leq1$ and positive integers $n$, $m$, and $j$ with $j\leq m+n$.

The probability that the collision involved $n$ particles is given by $\left(1-\delta\right)\left(\delta^{n}\right)$ and the probability that collision involved $m$ particles is given by $\left(1-\delta\right)\left(\delta^{m}\right)$, for a positive real number $\delta<1$. So I believe that the probability of forming $j$ particles in the collision is given by

$$Pr\left(j\right)=\left(1-\delta\right)^{2}\sum_{n=0}^{\infty}\delta^{n}\sum_{m=0}^{\infty}\delta^{m}R\left(n,m,j\right)$$

Question: Is there a simple form for $Pr\left(j\right)$? One that does not depend on $n$ or $m$?