15 Hypergeometric Function Properties 15.7 Continued Fractions 15.9 Relations to Other Functions

§15.8 Transformations of Variable

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Permalink:: http://dlmf.nist.gov/15.8
See also:: Annotations for Ch.15

§15.8(i) Linear Transformations
§15.8(ii) Linear Transformations: Limiting Cases
§15.8(iii) Quadratic Transformations
§15.8(iv) Quadratic Transformations (Continued)
§15.8(v) Cubic Transformations

§15.8(i) Linear Transformations

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Keywords:: hypergeometric function, linear, transformation of variable
A&S Ref:: 15.3
Notes:: For (15.8.1)–(15.8.4) see Olver (1997b, Chapter 5, §10). For (15.8.5) combine (15.8.1) and (15.8.2).
Referenced by:: §15.12(iii), §15.19(i), Erratum (V1.0.1) for Clarifications
Permalink:: http://dlmf.nist.gov/15.8.i
Amendment (effective with 1.0.1):: A sentence was added at the end of this subsection to identify alternative versions of the transformations (15.8.2)–(15.8.5).
See also:: Annotations for §15.8 and Ch.15

All functions in this subsection and §15.8(ii) assume their principal values.

15.8.1 $\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c}% ;\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}% \right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c};z\right),$ $|\operatorname{ph}\left(1-z\right)|<\pi$ .

15.8.2 $\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=% \frac{(-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({a% ,a-c+1\atop a-b+1};\frac{1}{z}\right)-\frac{(-z)^{-b}}{\Gamma\left(a\right)% \Gamma\left(c-b\right)}\mathbf{F}\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right),$ $|\operatorname{ph}\left(-z\right)|<\pi$ .

15.8.3 $\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=% \frac{(1-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}\mathbf{F}\left({% a,c-b\atop a-b+1};\frac{1}{1-z}\right)-\frac{(1-z)^{-b}}{\Gamma\left(a\right)% \Gamma\left(c-b\right)}\mathbf{F}\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right),$ $|\operatorname{ph}\left(-z\right)|<\pi$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\sin\NVar{z}$ : sine function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
Referenced by:: §15.8(i), §15.8(ii)
Permalink:: http://dlmf.nist.gov/15.8.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(i), §15.8 and Ch.15

15.8.4 $\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=% \frac{1}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({a,b% \atop a+b-c+1};1-z\right)-\frac{(1-z)^{c-a-b}}{\Gamma\left(a\right)\Gamma\left% (b\right)}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-z\right),$ $|\operatorname{ph}z|<\pi$ , $|\operatorname{ph}\left(1-z\right)|<\pi$ .

15.8.5 $\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c};z\right)=% \frac{z^{-a}}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({a,% a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)-\frac{(1-z)^{c-a-b}z^{a-c}}{\Gamma% \left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,1-a\atop c-a-b+1};1-% \frac{1}{z}\right),$ $|\operatorname{ph}z|<\pi$ , $|\operatorname{ph}\left(1-z\right)|<\pi$ .

For an alternative version of the transformations (15.8.2) and (15.8.3), see (15.10.25), and for an alternative version of the transformations (15.8.4) and (15.8.5), see (15.10.21).

§15.8(ii) Linear Transformations: Limiting Cases

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Keywords:: hypergeometric function, linear, transformation of variable
A&S Ref:: 15.3
Notes:: See Erdélyi et al. (1953a, §§2.1.4 and 2.3.1). (15.8.6) and (15.8.7) are obtained as limits of (15.8.2)–(15.8.5) as $a\to-m$ , together with (5.5.3). (15.8.9) and (15.8.11) follow from (15.8.10) and (15.8.8), respectively, via (15.8.1).
Referenced by:: §15.8(i)
Permalink:: http://dlmf.nist.gov/15.8.ii
See also:: Annotations for §15.8 and Ch.15

With $m=0,1,2,\dots$ , polynomial cases of (15.8.2)–(15.8.5) are given by

15.8.6 $F\left({-m,b\atop c};z\right)=\frac{{\left(b\right)_{m}}}{{\left(c\right)_{m}}% }(-z)^{m}F\left({-m,1-c-m\atop 1-b-m};\frac{1}{z}\right)=\frac{{\left(b\right)% _{m}}}{{\left(c\right)_{m}}}(1-z)^{m}F\left({-m,c-b\atop 1-b-m};\frac{1}{1-z}% \right),$

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : complex variable, : real or complex parameter, : real or complex parameter and : integer
Referenced by:: §15.8(ii), §15.8(ii), §18.17(vii)
Permalink:: http://dlmf.nist.gov/15.8.E6
Encodings:: TeX, pMML, png
Correction (effective with 1.1.2):: Pochhammer symbols now link to their definition.
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

15.8.7 $F\left({-m,b\atop c};z\right)=\frac{{\left(c-b\right)_{m}}}{{\left(c\right)_{m% }}}F\left({-m,b\atop b-c-m+1};1-z\right)=\frac{{\left(c-b\right)_{m}}}{{\left(% c\right)_{m}}}z^{m}F\left({-m,1-c-m\atop b-c-m+1};1-\frac{1}{z}\right),$

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), : complex variable, : real or complex parameter, : real or complex parameter and : integer
Referenced by:: §15.19(iv), §15.8(ii), §15.8(ii), (18.35.4_5)
Permalink:: http://dlmf.nist.gov/15.8.E7
Encodings:: TeX, pMML, png
Correction (effective with 1.1.2):: Pochhammer symbols now link to their definition.
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

with the understanding that if $b=-\ell$ , $\ell=0,1,2,\dots$ , then $m\leq\ell$ .

When is an integer limits are taken in (15.8.2) and (15.8.3) as follows.

If is a nonnegative integer, then

15.8.8 $\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(-z)^{-a}}{\Gamma\left(a+m\right% )}\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}(m-k-1)!}{k!\Gamma\left(c-a-k% \right)}z^{-k}+\frac{(-z)^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{% {\left(a+m\right)_{k}}}{k!(k+m)!\Gamma\left(c-a-k-m\right)}(-1)^{k}z^{-k-m}\*% \left(\ln\left(-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left% (a+k+m\right)-\psi\left(c-a-k-m\right)\right),$ $|z|>1,|\operatorname{ph}\left(-z\right)|<\pi$ ,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer
Referenced by:: §15.12(i), §15.19(iv), §15.8(ii), §15.8(ii), §16.11(i)
Permalink:: http://dlmf.nist.gov/15.8.E8
Encodings:: TeX, pMML, png
Correction (effective with 1.1.2):: Pochhammer symbols now link to their definition.
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

15.8.9 $\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(1-z)^{-a}}{\Gamma\left(a+m% \right)\Gamma\left(c-a\right)}\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}{\left% (c-a-m\right)_{k}}(m-k-1)!}{k!}(z-1)^{-k}+\frac{(-1)^{m}(1-z)^{-a-m}}{\Gamma% \left(a\right)\Gamma\left(c-a-m\right)}\sum_{k=0}^{\infty}\frac{{\left(a+m% \right)_{k}}{\left(c-a\right)_{k}}}{k!(k+m)!}(1-z)^{-k}\*(\ln\left(1-z\right)+% \psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left(a+k+m\right)-\psi\left(c% -a+k\right)),$ $|z-1|>1,|\operatorname{ph}\left(1-z\right)|<\pi$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer
Referenced by:: §15.8(ii)
Permalink:: http://dlmf.nist.gov/15.8.E9
Encodings:: TeX, pMML, png
Correction (effective with 1.1.2):: Pochhammer symbols now link to their definition.
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

In (15.8.8) when is a nonpositive integer $\ifrac{\psi\left(c-a-k-m\right)}{\Gamma\left(c-a-k-m\right)}$ is interpreted as $(-1)^{m+k+a-c+1}(m+k+a-c)!$ . Also, if is a nonpositive integer, then (15.8.6) applies.

Alternatively, if is a negative integer, then we interchange and in $\mathbf{F}\left(a,b;c;z\right)$ .

In a similar way, when is an integer limits are taken in (15.8.4) and (15.8.5) as follows.

If is a nonnegative integer, then

15.8.10 $\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{1}{\Gamma\left(a+m\right)% \Gamma\left(b+m\right)}\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}(m-k-1)!}{k!}(z-1)^{k}-\frac{(z-1)^{m}}{\Gamma\left(a\right)\Gamma% \left(b\right)}\sum_{k=0}^{\infty}\frac{{\left(a+m\right)_{k}}{\left(b+m\right% )_{k}}}{k!(k+m)!}(1-z)^{k}\*\left(\ln\left(1-z\right)-\psi\left(k+1\right)-% \psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b+k+m\right)\right),$ $|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi$ ,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer
Referenced by:: §15.4(ii), §15.8(ii)
Permalink:: http://dlmf.nist.gov/15.8.E10
Encodings:: TeX, pMML, png
Correction (effective with 1.1.2):: Pochhammer symbols now link to their definition.
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

15.8.11 $\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{z^{-a}}{\Gamma\left(a+m\right)% }\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}(m-k-1)!}{k!\Gamma\left(b+m-k\right% )}\left(1-\frac{1}{z}\right)^{k}-\frac{z^{-a}}{\Gamma\left(a\right)}\sum_{k=0}% ^{\infty}\frac{{\left(a+m\right)_{k}}}{k!(k+m)!\Gamma\left(b-k\right)}(-1)^{k}% \left(1-\frac{1}{z}\right)^{k+m}\*\left(\ln\left(\frac{1-z}{z}\right)-\psi% \left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b-k% \right)\right),$ $\Re z>\tfrac{1}{2},|\operatorname{ph}z|<\pi,|\operatorname{ph}\left(1-z\right)% |<\pi$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter, : integer and : integer
Referenced by:: §15.8(ii), §15.8(ii)
Permalink:: http://dlmf.nist.gov/15.8.E11
Encodings:: TeX, pMML, png
Correction (effective with 1.1.2):: Pochhammer symbols now link to their definition.
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

In (15.8.11) when is a nonpositive integer, $\ifrac{\psi\left(b-k\right)}{\Gamma\left(b-k\right)}$ is interpreted as $(-1)^{k-b+1}(k-b)!$ . Also, if or or both are nonpositive integers, then (15.8.7) applies.

Lastly, if is a negative integer, then we first apply the transformation

15.8.12 $\mathbf{F}\left(a,b;a+b-m;z\right)=(1-z)^{-m}\mathbf{F}\left(\tilde{a},\tilde{% b};\tilde{a}+\tilde{b}+m;z\right),$ $\tilde{a}=a-m,\tilde{b}=b-m$ .

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Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : integer
Permalink:: http://dlmf.nist.gov/15.8.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

§15.8(iii) Quadratic Transformations

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Keywords:: hypergeometric function, quadratic, transformation of variable
A&S Ref:: 15.3
Notes:: See Andrews et al. (1999, §3.1).
Referenced by:: §14.3(iii), §15.17(i), §15.9(iv), §31.7(i)
Permalink:: http://dlmf.nist.gov/15.8.iii
See also:: Annotations for §15.8 and Ch.15

A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation.

A necessary and sufficient condition that there exists a quadratic transformation is that at least one of the equations shown in Table 15.8.1 is satisfied.

Table 15.8.1: Quadratic transformations of the hypergeometric function.

Group 2	Group 3	Group 4
	$a=b+\frac{1}{2}$
	$b=a+\frac{1}{2}$	$c=\frac{1}{2}$
$c=\frac{1}{2}(a+b+1)$	$c=a+b+\frac{1}{2}$	$c=\frac{3}{2}$
	$c=a+b-\frac{1}{2}$

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Symbols:: : real or complex parameter, : real or complex parameter and : real or complex parameter
Referenced by:: §15.8(iii), §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.T1
See also:: Annotations for §15.8(iii), §15.8 and Ch.15

The hypergeometric functions that correspond to Groups 1 and 2 have as variable. The hypergeometric functions that correspond to Groups 3 and 4 have a nonlinear function of as variable. The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1).

In the equations that follow in this subsection all functions take their principal values.

Group 1 $\longrightarrow$ Group 3

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See also:: Annotations for §15.8(iii), §15.8 and Ch.15

15.8.13 $F\left({a,b\atop 2b};z\right)=\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{% 1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop b+\tfrac{1}{2}};\left(\frac{z}{2-z}% \right)^{2}\right),$ $|\operatorname{ph}\left(1-z\right)|<\pi$ ,

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.14 $F\left({a,b\atop 2b};z\right)=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1% }{2}a,b-\tfrac{1}{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right),$ $|\operatorname{ph}\left(1-z\right)|<\pi$ .

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

Group 2 $\longrightarrow$ Group 3

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See also:: Annotations for §15.8(iii), §15.8 and Ch.15

15.8.15 $F\left({a,b\atop a-b+1};z\right)=(1+z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+% \frac{1}{2}\atop a-b+1};\frac{4z}{(1+z)^{2}}\right),$

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(i), §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.16 $F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a-b% +\frac{1}{2}\atop a-b+1};\frac{-4z}{(1-z)^{2}}\right),$

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.17 $F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=(1-2z)^{-a}F\left({\frac{1}{2}a,% \frac{1}{2}a+\frac{1}{2}\atop\frac{1}{2}(a+b+1)};\frac{4z(z-1)}{(1-2z)^{2}}% \right),$ $\Re z<\frac{1}{2}$ ,

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\Re$ : real part, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.18 $F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=F\left({\frac{1}{2}a,\frac{1}{2}b% \atop\frac{1}{2}(a+b+1)};4z(1-z)\right),$ $\Re z<\frac{1}{2}$ .

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Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\Re$ : real part, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §8.17(i)
Permalink:: http://dlmf.nist.gov/15.8.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.19 $F\left({a,1-a\atop c};z\right)=(1-2z)^{1-a-c}(1-z)^{c-1}F\left({\frac{1}{2}(a+% c),\frac{1}{2}(a+c-1)\atop c};\frac{4z(z-1)}{(1-2z)^{2}}\right),$ $\Re z<\frac{1}{2}$ ,

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\Re$ : real part, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.20 $F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}F\left({\frac{1}{2}(c-a),\frac{1}{2}% (a+c-1)\atop c};4z(1-z)\right),$ $\Re z<\frac{1}{2}$ .

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\Re$ : real part, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

Group 2 $\longrightarrow$ Group 1

ⓘ

See also:: Annotations for §15.8(iii), §15.8 and Ch.15

15.8.21 $F\left({a,b\atop a-b+1};z\right)=\left(1+\sqrt{z}\right)^{-2a}F\left({a,a-b+% \tfrac{1}{2}\atop 2a-2b+1};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right),$ $|\operatorname{ph}z|<\pi$ ,

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.8(iv)
Permalink:: http://dlmf.nist.gov/15.8.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.22 $F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(\frac{\sqrt{1-z^{-1}}-1}{% \sqrt{1-z^{-1}}+1}\right)^{a}F\left({a,\tfrac{1}{2}(a+b)\atop a+b};\frac{4% \sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}\right),$ $|\operatorname{ph}\left(-z\right)|<\pi$ , $\Re z<\frac{1}{2}$ .

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.23 $F\left({a,1-a\atop c};z\right)=\left(\sqrt{1-z^{-1}}-1\right)^{1-a}\left(\sqrt% {1-z^{-1}}+1\right)^{a-2c+1}\left(1-z^{-1}\right)^{c-1}F\left({c-a,c-\tfrac{1}% {2}\atop 2c-1};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{-1}}+1\right)^{2}}% \right),$ $|\operatorname{ph}\left(-z\right)|<\pi$ , $\Re z<\frac{1}{2}$ .

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $\Re$ : real part, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

Group 2 $\longrightarrow$ Group 4

ⓘ

See also:: Annotations for §15.8(iii), §15.8 and Ch.15

15.8.24 $F\left({a,b\atop a-b+1};z\right)=(1-z)^{-a}\frac{\Gamma\left(a-b+1\right)% \Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)% \Gamma\left(\tfrac{1}{2}a-b+1\right)}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a-b+% \tfrac{1}{2}\atop\tfrac{1}{2}};\left(\frac{z+1}{z-1}\right)^{2}\right)+(1+z)(1% -z)^{-a-1}\frac{\Gamma\left(a-b+1\right)\Gamma\left(-\tfrac{1}{2}\right)}{% \Gamma\left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}a-b+\tfrac{1}{2}\right% )}F\left({\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}a-b+1\atop\tfrac{3}{2}};\left% (\frac{z+1}{z-1}\right)^{2}\right),$ $|\operatorname{ph}\left(-z\right)|<\pi$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.8.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.25 $F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\frac{\Gamma\left(\tfrac{1}{2}(a% +b+1)\right)\Gamma\left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}a+\tfrac{% 1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}F\left({\tfrac{1}{2}% a,\tfrac{1}{2}b\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)\frac{\Gamma\left(% \tfrac{1}{2}(a+b+1)\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma\left(\tfrac% {1}{2}a\right)\Gamma\left(\tfrac{1}{2}b\right)}F\left({\tfrac{1}{2}a+\tfrac{1}% {2},\tfrac{1}{2}b+\tfrac{1}{2}\atop\tfrac{3}{2}};(1-2z)^{2}\right),$ $|\operatorname{ph}z|<\pi$ , $|\operatorname{ph}\left(1-z\right)|<\pi$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.26 $F\left({a,1-a\atop c};z\right)=(1-z)^{c-1}\frac{\Gamma\left(c\right)\Gamma% \left(\tfrac{1}{2}\right)}{\Gamma\left(\tfrac{1}{2}(c-a+1)\right)\Gamma\left(% \tfrac{1}{2}c+\tfrac{1}{2}a\right)}F\left({\tfrac{1}{2}c-\tfrac{1}{2}a,\tfrac{% 1}{2}c+\tfrac{1}{2}a-\tfrac{1}{2}\atop\tfrac{1}{2}};(1-2z)^{2}\right)+(1-2z)(1% -z)^{c-1}\frac{\Gamma\left(c\right)\Gamma\left(-\tfrac{1}{2}\right)}{\Gamma% \left(\tfrac{1}{2}c-\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}(c+a-1)\right)% }F\left({\tfrac{1}{2}c-\tfrac{1}{2}a+\tfrac{1}{2},\tfrac{1}{2}c+\tfrac{1}{2}a% \atop\tfrac{3}{2}};(1-2z)^{2}\right),$ $|\operatorname{ph}z|<\pi$ , $|\operatorname{ph}\left(1-z\right)|<\pi$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

Group 4 $\longrightarrow$ Group 2

ⓘ

Keywords:: hypergeometric function, quadratic, transformation of variable
See also:: Annotations for §15.8(iii), §15.8 and Ch.15

15.8.27 $\frac{2\Gamma\left(\tfrac{1}{2}\right)\Gamma\left(a+b+\tfrac{1}{2}\right)}{% \Gamma\left(a+\tfrac{1}{2}\right)\Gamma\left(b+\tfrac{1}{2}\right)}F\left(a,b;% \tfrac{1}{2};z\right)=F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}-\tfrac{1}{2}% \sqrt{z}\right)+F\left(2a,2b;a+b+\tfrac{1}{2};\tfrac{1}{2}+\tfrac{1}{2}\sqrt{z% }\right),$ $|\operatorname{ph}z|<\pi$ , $|\operatorname{ph}\left(1-z\right)|<\pi$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.8.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

15.8.28 $\frac{2\sqrt{z}\Gamma\left(-\tfrac{1}{2}\right)\Gamma\left(a+b-\tfrac{1}{2}% \right)}{\Gamma\left(a-\tfrac{1}{2}\right)\Gamma\left(b-\tfrac{1}{2}\right)}F% \left(a,b;\tfrac{3}{2};z\right)=F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}% -\tfrac{1}{2}\sqrt{z}\right)-F\left(2a-1,2b-1;a+b-\tfrac{1}{2};\tfrac{1}{2}+% \tfrac{1}{2}\sqrt{z}\right),$ $|\operatorname{ph}z|<\pi$ , $|\operatorname{ph}\left(1-z\right)|<\pi$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.8.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iii), §15.8(iii), §15.8 and Ch.15

§15.8(iv) Quadratic Transformations (Continued)

ⓘ

Permalink:: http://dlmf.nist.gov/15.8.iv
See also:: Annotations for §15.8 and Ch.15

When the intersection of two groups in Table 15.8.1 is not empty there exist special quadratic transformations, with only one free parameter, between two hypergeometric functions in the same group.

Examples

ⓘ

See also:: Annotations for §15.8(iv), §15.8 and Ch.15

$b=\tfrac{1}{3}a+\tfrac{1}{3}$ , in Groups 1 and 2.

(15.8.21) becomes

15.8.29 $F\left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=% \left(1+\sqrt{z}\right)^{-2a}\*F\left({a,\tfrac{2}{3}a+\tfrac{1}{6}\atop\tfrac% {4}{3}a+\tfrac{1}{3}};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right).$

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, : complex variable and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iv), §15.8(iv), §15.8 and Ch.15

This is a quadratic transformation between two cases in Group 1.

We can also use (15.8.13), followed by the inverse of (15.8.15), and obtain

15.8.30 $\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}% {2}\atop\tfrac{1}{3}a+\tfrac{5}{6}};\left(\frac{z}{2-z}\right)^{2}\right)=F% \left({a,\tfrac{1}{3}a+\tfrac{1}{3}\atop\tfrac{2}{3}a+\tfrac{2}{3}};z\right)=(% 1+z)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}{2}a+\tfrac{1}{2}\atop\tfrac{2}{3}a+% \tfrac{2}{3}};\frac{4z}{(1+z)^{2}}\right),$

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.8.E30
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(iv), §15.8(iv), §15.8 and Ch.15

which is a quadratic transformation between two cases in Group 3.

For further examples see Andrews et al. (1999, pp. 130–132 and 176–177).

§15.8(v) Cubic Transformations

ⓘ

Keywords:: cubic, hypergeometric function, transformation of variable
Notes:: See Goursat (1881, Eq. (110)) and Watson (1910). The version of (15.8.31) given in Erdélyi et al. (1953a, p. 114 (40)) contains a typographical error. For (15.8.33) see Chan (1998).
Referenced by:: §31.7(i), Erratum (V1.0.10) for References
Permalink:: http://dlmf.nist.gov/15.8.v
Addition (effective with 1.0.10):: The reference to Tu and Yang (2013) was added at the end of this subsection.
See also:: Annotations for §15.8 and Ch.15

Examples

ⓘ

See also:: Annotations for §15.8(v), §15.8 and Ch.15

15.8.31 $F\left({3a,3a+\frac{1}{2}\atop 4a+\frac{2}{3}};z\right)=\left(1-\tfrac{9}{8}z% \right)^{-2a}\*F\left({a,a+\frac{1}{2}\atop 2a+\frac{5}{6}};\frac{27z^{2}(z-1)% }{(9z-8)^{2}}\right),$ $\Re z<\frac{8}{9}$ .

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\Re$ : real part, : complex variable and : real or complex parameter
Referenced by:: §15.8(v)
Permalink:: http://dlmf.nist.gov/15.8.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(v), §15.8(v), §15.8 and Ch.15

With $\zeta={\mathrm{e}}^{\ifrac{2\pi\mathrm{i}}{3}}(1-z)/\left(z-{\mathrm{e}}^{% \ifrac{4\pi\mathrm{i}}{3}}\right)$

15.8.32 $\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\Gamma% \left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}\right)}F\left({a,a+\frac{1}{% 3}\atop\frac{2}{3}};z^{-3}\right)+\frac{{\mathrm{e}}^{\frac{1}{3}\pi\mathrm{i}% }}{z\Gamma\left(a\right)\Gamma\left(\frac{4}{3}\right)}F\left({a+\frac{1}{3},a% +\frac{2}{3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac{3^{\frac{3}{2}a+\frac% {1}{2}}{\mathrm{e}}^{\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{1}{3}\right% )(1-\zeta)^{a}}{2\pi\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}F\left({a+% \frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{-1}\right),$

, $|\operatorname{ph}\left(-z\right)|<\frac{1}{3}\pi$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, : complex variable, : real or complex parameter and $\zeta$ : change of variable
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.8.E32
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(v), §15.8(v), §15.8 and Ch.15

Ramanujan’s Cubic Transformation

ⓘ

Keywords:: Ramanujan’s cubic transformation, cubic, hypergeometric function, transformation of variable
See also:: Annotations for §15.8(v), §15.8 and Ch.15

15.8.33 $F\left({\frac{1}{3},\frac{2}{3}\atop 1};1-\left(\frac{1-z}{1+2z}\right)^{3}% \right)=(1+2z)F\left({\frac{1}{3},\frac{2}{3}\atop 1};z^{3}\right),$

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function and : complex variable
Referenced by:: §15.8(v)
Permalink:: http://dlmf.nist.gov/15.8.E33
Encodings:: TeX, pMML, png
See also:: Annotations for §15.8(v), §15.8(v), §15.8 and Ch.15

provided that lies in the intersection of the open disks $\left|z-\frac{1}{4}\pm\frac{1}{4}\sqrt{3}\mathrm{i}\right|<\frac{1}{2}\sqrt{3}$ , or equivalently, $\left|\operatorname{ph}\left(\ifrac{(1-z)}{(1+2z)}\right)\right|<\pi/3$ . This is used in a cubic analog of the arithmetic-geometric mean. See Borwein and Borwein (1991), and also Berndt et al. (1995).

For further examples and higher-order transformations see Goursat (1881), Watson (1910), Vidūnas (2005), and Tu and Yang (2013); see also Erdélyi et al. (1953a, pp. 67 and 113–114).

15.7 Continued Fractions 15.9 Relations to Other Functions

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§15.8 Transformations of Variable

Contents

§15.8(i) Linear Transformations

§15.8(ii) Linear Transformations: Limiting Cases

§15.8(iii) Quadratic Transformations

Group 1 Group 3

Group 2 Group 3

Group 2 Group 1

Group 2 Group 4

Group 4 Group 2

§15.8(iv) Quadratic Transformations (Continued)

Examples

§15.8(v) Cubic Transformations

Examples

Ramanujan’s Cubic Transformation

Group 1 $\longrightarrow$ Group 3

Group 2 $\longrightarrow$ Group 3

Group 2 $\longrightarrow$ Group 1

Group 2 $\longrightarrow$ Group 4

Group 4 $\longrightarrow$ Group 2