Background
For the past week, I have struggled with a complicated probability distribution in Mathematica. I want to analytically show that it is normalized, or at least, converges. This is the last part of a project in which I have finished all numerical verifications.
I have rewritten the probability distribution as nicely as I could. I would be grateful if anyone here can help me handle the 2F1 expressions. Here it is:
$$\mathrm{P}(L) = f(L)[g_1(L)+g_2(L)+g_3(L)+g_4(L)]$$ where \begin{align*} f(L) &= \dfrac{a^L}{(a-b)^{C}~(C-1)!~L!} \\ g_1(L) &= \dfrac{(C-1)!~L! \left(b a^C+d (a-b)^C\right)}{(a+b+d)^{L+1}} \\ g_2(L) &= \frac{d ~(C+L)! ~(a-b)^C \, _2F_1\left(L+1,C+L+1;L+2;-\frac{a+d}{a}\right)}{(L+1)~a^{L+1}} \\ g_3(L) &= -\frac{d ~(C+L)! ~(a-b)^C \, _2F_1\left(L+1,C+L+1;L+2;-\frac{a+b+d}{a}\right)}{(L+1)~a^{L+1}} \\ g_4(L) &= -\frac{b (C+L)! (a-b)^C \left(\frac{a}{a-b}\right)^{C+L+1} \, _2F_1\left(L+1,C+L+1;L+2;-\frac{a+b+d}{a-b}\right)}{(L+1)~a^{L+1}} \end{align*}
where $a,b,d$ are positive real numbers and $C,L$ are nonnegative integers.
Goal
- Showing that $\sum_{L=0}^{\infty}P(L) = 1.$ If not possible, I would like to show the sum is convergent.
Attempts
Notice the similarities between the $_2F_1$ expressions in $g_2,g_3,g_4.$ I applied some transformations on them, but was still unable to exploit them in Mathematica. Here are the transformations I tried, in all combinations: \begin{align} _2F_1(a,b;c;z) &= (1-z)^{c-a-b} \cdot {}_2F_1\left(c-a,c-b;c;z\right) \\ _2F_1(a,b;c;z) &= (1-z)^{-a} \cdot {}_2F_1\left(a,c-b;c;\frac{z}{(z-1)}\right) \\ _2F_1(1,b;c;z) &= (c-1)z^{1-c}(1-z)^{c-b-1}B_z\left(c-1,b-c+1\right) \end{align} where $B_z(a,b)$ denotes the incomplete Beta function.
Whenever I got an incomplete Beta function in the expression, however, I had $a>0$, $b<0$, and $z<-1$. This was not easy to understand.
My main issue is with $g_2,g_3,g_4$. Analytical insight would be highly appreciated. Maple or Mathematica code is also welcome. Thank you.