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
Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m
2F1(1,1;2;z)=-log(1-z)/z
2F1(1/2,1;3/2;z^2)=log((1+z)/(1-z))/(2z)
2F1(1/2,1;3/2;-z^2)=arctan(z)/z
2F1(1/2,1/2;3/2;z^2)=arcsin(z)/z
Sqrt(1-z^2)2F1(1,1;3/2;z^2)=arcsin(z)/z
z2F1(1,1;2,-z) equals log(1+z)
Relationship between Chebyshev T and hypergeometric 2F1
Relationship between Chebyshev U and hypergeometric 2F1
Relationship between Legendre polynomial and hypergeometric 2F1
Relationship between incomplete beta and hypergeometric 2F1
2F1(a,b;a+b+1/2;z)^2=3F2(2a,a+b,2b;a+b+1/2,2a+2b;z)
0F1(;r;z)0F1(;s;z)=2F1(r/2+s/2, r/2+s/2-1/2;r,s,r+s-1;4z)
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
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 2.8 (1)$
- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 5.1 (1)$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $15.1.1$
- 1964: W.N. Bailey: Generalized Hypergeometric Series ... (previous) ... (next): Section $1.1$
