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

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 2.8 (1)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $15.1.1$