# Hypergeometric 2F1

The (Gauss) hypergeometric ${}_2F_1$ function (often written simply as $F$) is defined by the series $${}_2F_1(a,b;c;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a)_k (b)_k}{(c)_k} \dfrac{z^k}{k!},$$ where $(a)_k$ denotes the Pochhammer symbol and $c \neq 0, -1, -2, \ldots$. It is a special case of the hypergeometric pFq function.

# Properties

## Contiguous relations

We adopt the following notations: $$F = {}_2F_1(a,b;c;z),$$ $$F(a \pm 1)={}_2F_1(a \pm 1,b;c;z),$$ $$F(b \pm 1)={}_2F_1(a, b\pm 1;c;z),$$ and $$F(c \pm 1)={}_2F_1(a,b;c \pm 1;z).$$ (c-2a-(b-a)z)2F1+a(1-z)2F1(a+1)-(c-a)2F1(a-1)=0
(b-a)2F1+a2F1(a+1)-b2F1(b+1)=0
(c-a-b)2F1+a(1-z)2F1(a+1)-(c-b)2F1(b-1)=0
c(a-(c-b)z)2F1-ac(1-z)2F1(a+1)+(c-a)(c-b)z2F1(c+1)=0
(c-a-1)2F1+a2F1(a+1)-(c-1)2F1(c-1)=0
(c-a-b)2F1-(c-a)2F1(a-1)+b(1-z)2F1(b+1)=0
(b-a)(1-z)2F1-(c-a)2F1(a-1)+(c-b)2F1(b-1)=0
c(1-z)2F1-c2F1(a-1)+(c-b)z2F1(c+1)=0
(a-1-(c-b-1)z)2F1+(c-a)2F1(a-1)-(c-1)(1-z)2F1(c-1)=0
(c-2b+(b-a)z)2F1+b(1-z)2F1(b+1)-(c-b)2F1(b-1)=0
c(b-(c-a)z)2F1-bc(1-z)2F1(b+1)+(c-a)(c-b)z2F1(c+1)=0
(c-b-1)2F1+b2F1(b+1)-(c-1)2F1(c-1)=0
c(1-z)2F1-c2F1(b-1)+(c-a)2F1(c+1)=0
(b-1-(c-a-1)z)2F1+(c-b)2F1(b-1)-(c-1)(1-z)2F1(c-1)=0
c(c-1-(2c-a-b-1)z)2F1+(c-a)(c-b)z2F1(c+1)-c(c-1)(1-z)2F1(c-1)=0

# References

Hypergeometric functions