How to derive Such infinite sum representation for Hypergeometric function?

Question

I was reading a paper $[1]$ in which authors claimed that we can simplify below Gauss function to finite series if $m $ and $v$ are positive integers.

$$ _2F_{1}(v,m+v;m+1;x)=\psi\sum_{c=0}^{v-1} {v+m-1\choose c}{2v-2-c\choose v-1} \gamma^c$$

for values of $\psi$ and the$\gamma$ include x values as beow:

$$\gamma = \frac{1-x}{x}$$

However, I know $[2]$ the Hyper geometric, $_2F_1(\alpha,\beta;z;x)$,function terminates if $\alpha$ and $\beta$ is non-positive integers. This bothers me how such positive arguments can lead to a finite series representation.

Thanks

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References:

[1] https://ieeexplore.ieee.org/document/1247815

[2] Table of Integrals, Series and Products, I.S, Gradshteyn, et. al., 2007.

Answer 1

If you look at DLMF 15.8.1 you will see that:
$F\left(\begin{array}{c} a,b\\ c \end{array};z\right)=\frac{1}{\left(1-x\right)^{a}}F\left(\begin{array}{c} a,c-b\\ c \end{array};\frac{z}{z-1}\right)$
Now in your case $b=m+v,c=m+1$
So $c-b=m+1-m-v=1-v$
and if $v\in\mathcal{N}$
$-v$ is a negative integer.