Generalised Binomial Series

Question

I am reading an old paper "1971A Generalized Negative Binomial Distribution" by G. C. Jain and P. C. Consul. The paper mainly start from $$(1+z)^n=\sum_{x=0}^{\infty}\frac{n}{n+x\beta}\binom{n+x\beta}{x}z^x(1+z)^{-\beta x},\quad where \quad |\frac{\beta z}{1+z}|<1.$$

I can understand this. But I cannot understand why (the last two lines of the paper) $$\sum_{x=0}^{\infty}\frac{n}{n+\beta x}\binom{n+\beta x}{x}=1.$$

Expecting some hints from you.

Thanks in advance.

Answer 1

I think this result provided by the paper is wrong. Simplify run the numerical calculation in any mathematical software will show this is not a probability distribution function at all.

The range of $\beta$, $\beta=0$ or $\beta \geq 1$, could be found in "1980The Generalized Negative Binomial Distribution and its Characterization by ZeroRegression". There are several subsequent papers pointing out the parameter setting in 1971 G. C. Jain and P. C. Consul is not correct.

Let me know if you have other comments.