# On Hyper-geometric Functions and its recurrence relation

I research in generating functions of Hyper-geometric functions $_2F_1(a+n,b;c+n;x)$ using Lie group theoretic method and so the recurrence relation is important in this method.

I want recurrence relation like these in the image

the author write that Slater in "Generalized Hypergeometric Functions" give this relations for $_2F_1(a,b;c+n;x)$

But , I don't know how to find this recurrence relation for another $_2F_1(a+n,b;c+n;x)$

Update:- Are these recurrence relation correct ? $$[z \frac{\partial}{\partial z}+c+n-1] _2F_1 (a+n,b;c+n;z)=(c+n-1) _2F_1 (a+n,b;c+n-1;z)$$

and

$$[(c+n)(1-z) \frac{\partial}{\partial z}-(c+n)(a+b-c)] _2F_1 (a+n,b;c+n;z)=(c-a)(c+n-b)_2F_1 (a+n,b;c+n+1;z)$$

please, help me in this .

• $a$ is just a parameter; nothing will become invalid if you change it to $a+n$. – Dima Pasechnik Mar 5 '16 at 21:23
• do you mean that the recurrence relation in the image will not be change ? – Hamada Al Mar 5 '16 at 21:29
• right, just replace $a$ by $a+n$ – Dima Pasechnik Mar 5 '16 at 22:12
• please , see my update . – Hamada Al Mar 6 '16 at 13:15