Difference between revisions of "Hypergeometric 2F1"
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The (Gauss) hypergeometric ${}_2F_1$ function (often written simply as $F$) is defined by the series | The (Gauss) hypergeometric ${}_2F_1$ function (often written simply as $F$) is defined by the series | ||
$${}_2F_1(a,b;c;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a)_k (b)_k}{(c)_k} \dfrac{z^k}{k!},$$ | $${}_2F_1(a,b;c;z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{(a)_k (b)_k}{(c)_k} \dfrac{z^k}{k!},$$ | ||
| − | where $(a)_k$ denotes the [[Pochhammer]] symbol. It is a special case of the [[hypergeometric pFq]] function. | + | 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= | =Properties= | ||
| Line 18: | Line 18: | ||
=References= | =References= | ||
| + | * {{BookReference|Higher Transcendental Functions Volume I|1953|Harry Bateman|prev=Beta is symmetric|next=findme}}: $\S 2.8 (1)$ | ||
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m}}: $15.1.1$ | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=Limit of (1/Gamma(c))*2F1(a,b;c;z) as c approaches -m}}: $15.1.1$ | ||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Revision as of 02:41, 16 September 2016
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): $15.1.1$
