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

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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		$\mathbf{F}\left({a,b\atop c};z\right)=(1-z)^{-a}\mathbf{F}\left({a,c-b\atop c}% ;\frac{z}{z-1}\right)=(1-z)^{-b}\mathbf{F}\left({c-a,b\atop c};\frac{z}{z-1}% \right)=(1-z)^{c-a-b}\mathbf{F}\left({c-a,c-b\atop c};z\right),$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ Symbols: $\pi$ : 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.10(ii), §15.13, §15.16, (15.4.27), §15.4(i), §15.4(iii), §15.8(i), §15.8(ii), §15.8(iii), §8.17(i) Permalink: http://dlmf.nist.gov/15.8.E1 Encodings: TeX, pMML, png See also: Annotations for §15.8(i), §15.8 and Ch.15

15.8.2	$\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c}% ;z\right)$	$\displaystyle=\frac{(-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}% \mathbf{F}\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)-\frac{(-z)^{-b}}{% \Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,b-c+1\atop b-a+1% };\frac{1}{z}\right),$
$\|\operatorname{ph}\left(-z\right)\|<\pi$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.12(i), §15.16, §15.8(i), §15.8(i), §15.8(ii), §15.8(ii), §15.8(ii) Permalink: http://dlmf.nist.gov/15.8.E2 Encodings: TeX, pMML, png See also: Annotations for §15.8(i), §15.8 and Ch.15
15.8.3	$\displaystyle\frac{\sin\left(\pi(b-a)\right)}{\pi}\mathbf{F}\left({a,b\atop c}% ;z\right)$	$\displaystyle=\frac{(1-z)^{-a}}{\Gamma\left(b\right)\Gamma\left(c-a\right)}% \mathbf{F}\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)-\frac{(1-z)^{-b}}{% \Gamma\left(a\right)\Gamma\left(c-b\right)}\mathbf{F}\left({b,c-a\atop b-a+1};% \frac{1}{1-z}\right),$
$\|\operatorname{ph}\left(-z\right)\|<\pi$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : 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	$\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c% };z\right)$	$\displaystyle=\frac{1}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}\mathbf{F}% \left({a,b\atop a+b-c+1};1-z\right)-\frac{(1-z)^{c-a-b}}{\Gamma\left(a\right)% \Gamma\left(b\right)}\mathbf{F}\left({c-a,c-b\atop c-a-b+1};1-z\right),$
$\|\operatorname{ph}z\|<\pi$ , $\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.16, §15.19(iv), §15.2(i), §15.4(ii), §15.8(i), §15.8(i), §15.8(ii) Permalink: http://dlmf.nist.gov/15.8.E4 Encodings: TeX, pMML, png See also: Annotations for §15.8(i), §15.8 and Ch.15
15.8.5	$\displaystyle\frac{\sin\left(\pi(c-a-b)\right)}{\pi}\mathbf{F}\left({a,b\atop c% };z\right)$	$\displaystyle=\frac{z^{-a}}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}% \mathbf{F}\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)-\frac{(1-z)^{c-a-b% }z^{a-c}}{\Gamma\left(a\right)\Gamma\left(b\right)}\mathbf{F}\left({c-a,1-a% \atop c-a-b+1};1-\frac{1}{z}\right),$
$\|\operatorname{ph}z\|<\pi$ , $\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.8(i), §15.8(i), §15.8(ii), §15.8(ii), §15.8(ii) Permalink: http://dlmf.nist.gov/15.8.E5 Encodings: TeX, pMML, png See also: Annotations for §15.8(i), §15.8 and Ch.15

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	$\displaystyle F\left({-m,b\atop c};z\right)$	$\displaystyle=\frac{{\left(b\right)_{m}}}{{\left(c\right)_{m}}}(-z)^{m}F\left(% {-m,1-c-m\atop 1-b-m};\frac{1}{z}\right)=\frac{{\left(b\right)_{m}}}{{\left(c% \right)_{m}}}(1-z)^{m}F\left({-m,c-b\atop 1-b-m};\frac{1}{1-z}\right),$
ⓘ 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), $z$ : complex variable, $b$ : real or complex parameter, $c$ : real or complex parameter and $m$ : 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	$\displaystyle F\left({-m,b\atop c};z\right)$	$\displaystyle=\frac{{\left(c-b\right)_{m}}}{{\left(c\right)_{m}}}F\left({-m,b% \atop b-c-m+1};1-z\right)=\frac{{\left(c-b\right)_{m}}}{{\left(c\right)_{m}}}z% ^{m}F\left({-m,1-c-m\atop b-c-m+1};1-\frac{1}{z}\right),$
ⓘ 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), $z$ : complex variable, $b$ : real or complex parameter, $c$ : real or complex parameter and $m$ : 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 $b-a$ is an integer limits are taken in (15.8.2) and (15.8.3) as follows.

If $b-a$ is a nonnegative integer, then

15.8.8		$\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(-z)^{-a}}{\Gamma\left(a+m\right% )}\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}(m-k-1)!}{k!\Gamma\left(c-a-k% \right)}z^{-k}+\frac{(-z)^{-a}}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{% {\left(a+m\right)_{k}}}{k!(k+m)!\Gamma\left(c-a-k-m\right)}(-1)^{k}z^{-k-m}\*% \left(\ln\left(-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left% (a+k+m\right)-\psi\left(c-a-k-m\right)\right),$
$\|z\|>1,\|\operatorname{ph}\left(-z\right)\|<\pi$ ,
ⓘ 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 $n!$ ), $\ln\NVar{z}$ : 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, $z$ : complex variable, $a$ : real or complex parameter, $c$ : real or complex parameter, $k$ : integer and $m$ : 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		$\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(1-z)^{-a}}{\Gamma\left(a+m% \right)\Gamma\left(c-a\right)}\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}{\left% (c-a-m\right)_{k}}(m-k-1)!}{k!}(z-1)^{-k}+\frac{(-1)^{m}(1-z)^{-a-m}}{\Gamma% \left(a\right)\Gamma\left(c-a-m\right)}\sum_{k=0}^{\infty}\frac{{\left(a+m% \right)_{k}}{\left(c-a\right)_{k}}}{k!(k+m)!}(1-z)^{-k}\*(\ln\left(1-z\right)+% \psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left(a+k+m\right)-\psi\left(c% -a+k\right)),$
$\|z-1\|>1,\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ 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 $n!$ ), $\ln\NVar{z}$ : 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, $z$ : complex variable, $a$ : real or complex parameter, $c$ : real or complex parameter, $k$ : integer and $m$ : 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 $c-a-k-m$ 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 $a$ is a nonpositive integer, then (15.8.6) applies.

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

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

If $c-a-b$ is a nonnegative integer, then

15.8.10		$\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{1}{\Gamma\left(a+m\right)% \Gamma\left(b+m\right)}\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}{\left(b% \right)_{k}}(m-k-1)!}{k!}(z-1)^{k}-\frac{(z-1)^{m}}{\Gamma\left(a\right)\Gamma% \left(b\right)}\sum_{k=0}^{\infty}\frac{{\left(a+m\right)_{k}}{\left(b+m\right% )_{k}}}{k!(k+m)!}(1-z)^{k}\*\left(\ln\left(1-z\right)-\psi\left(k+1\right)-% \psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b+k+m\right)\right),$
$\|z-1\|<1,\|\operatorname{ph}\left(1-z\right)\|<\pi$ ,
ⓘ 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 $n!$ ), $\ln\NVar{z}$ : 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $k$ : integer and $m$ : 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		$\mathbf{F}\left({a,b\atop a+b+m};z\right)=\frac{z^{-a}}{\Gamma\left(a+m\right)% }\sum_{k=0}^{m-1}\frac{{\left(a\right)_{k}}(m-k-1)!}{k!\Gamma\left(b+m-k\right% )}\left(1-\frac{1}{z}\right)^{k}-\frac{z^{-a}}{\Gamma\left(a\right)}\sum_{k=0}% ^{\infty}\frac{{\left(a+m\right)_{k}}}{k!(k+m)!\Gamma\left(b-k\right)}(-1)^{k}% \left(1-\frac{1}{z}\right)^{k+m}\*\left(\ln\left(\frac{1-z}{z}\right)-\psi% \left(k+1\right)-\psi\left(k+m+1\right)+\psi\left(a+k+m\right)+\psi\left(b-k% \right)\right),$
$\Re z>\tfrac{1}{2},\|\operatorname{ph}z\|<\pi,\|\operatorname{ph}\left(1-z\right)% \|<\pi$ .
ⓘ 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 $n!$ ), $\ln\NVar{z}$ : 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $k$ : integer and $m$ : 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 $b-k$ 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 $a$ or $b$ or both are nonpositive integers, then (15.8.7) applies.

Lastly, if $c-a-b$ is a negative integer, then we first apply the transformation

15.8.12		$\mathbf{F}\left(a,b;a+b-m;z\right)=(1-z)^{-m}\mathbf{F}\left(\tilde{a},\tilde{% b};\tilde{a}+\tilde{b}+m;z\right),$
$\tilde{a}=a-m,\tilde{b}=b-m$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $m$ : 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 1	Group 2	Group 3	Group 4
	$c=a-b+1$	$a=b+\frac{1}{2}$
$c=2a$	$c=b-a+1$	$b=a+\frac{1}{2}$	$c=\frac{1}{2}$
$c=2b$	$c=\frac{1}{2}(a+b+1)$	$c=a+b+\frac{1}{2}$	$c=\frac{3}{2}$
	$a+b=1$	$c=a+b-\frac{1}{2}$

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Symbols:: $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : 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 $z$ as variable. The hypergeometric functions that correspond to Groups 3 and 4 have a nonlinear function of $z$ 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	$\displaystyle F\left({a,b\atop 2b};z\right)$	$\displaystyle=\left(1-\tfrac{1}{2}z\right)^{-a}F\left({\tfrac{1}{2}a,\tfrac{1}% {2}a+\tfrac{1}{2}\atop b+\tfrac{1}{2}};\left(\frac{z}{2-z}\right)^{2}\right),$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ ,
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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	$\displaystyle F\left({a,b\atop 2b};z\right)$	$\displaystyle=\left(1-z\right)^{-\ifrac{a}{2}}F\left({\tfrac{1}{2}a,b-\tfrac{1% }{2}a\atop b+\tfrac{1}{2}};\frac{z^{2}}{4z-4}\right),$
$\|\operatorname{ph}\left(1-z\right)\|<\pi$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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

ⓘ

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

15.8.15	$\displaystyle F\left({a,b\atop a-b+1};z\right)$	$\displaystyle=(1+z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop a-b% +1};\frac{4z}{(1+z)^{2}}\right),$
$\|z\|<1$ ,
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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	$\displaystyle F\left({a,b\atop a-b+1};z\right)$	$\displaystyle=(1-z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a-b+\frac{1}{2}\atop a% -b+1};\frac{-4z}{(1-z)^{2}}\right),$
$\|z\|<1$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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	$\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)$	$\displaystyle=(1-2z)^{-a}F\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop% \frac{1}{2}(a+b+1)};\frac{4z(z-1)}{(1-2z)^{2}}\right),$
$\Re z<\frac{1}{2}$ ,
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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	$\displaystyle F\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)$	$\displaystyle=F\left({\frac{1}{2}a,\frac{1}{2}b\atop\frac{1}{2}(a+b+1)};4z(1-z% )\right),$
$\Re z<\frac{1}{2}$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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	$\displaystyle F\left({a,1-a\atop c};z\right)$	$\displaystyle=(1-2z)^{1-a-c}(1-z)^{c-1}F\left({\frac{1}{2}(a+c),\frac{1}{2}(a+% c-1)\atop c};\frac{4z(z-1)}{(1-2z)^{2}}\right),$
$\Re z<\frac{1}{2}$ ,
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $c$ : 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	$\displaystyle F\left({a,1-a\atop c};z\right)$	$\displaystyle=(1-z)^{c-1}F\left({\frac{1}{2}(c-a),\frac{1}{2}(a+c-1)\atop c};4% z(1-z)\right),$
$\Re z<\frac{1}{2}$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $c$ : 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	$\displaystyle F\left({a,b\atop a-b+1};z\right)$	$\displaystyle=\left(1+\sqrt{z}\right)^{-2a}F\left({a,a-b+\tfrac{1}{2}\atop 2a-% 2b+1};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right),$
$\|\operatorname{ph}z\|<\pi$ , $\|z\|<1$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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	$\displaystyle F\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)$	$\displaystyle=\left(\frac{\sqrt{1-z^{-1}}-1}{\sqrt{1-z^{-1}}+1}\right)^{a}F% \left({a,\tfrac{1}{2}(a+b)\atop a+b};\frac{4\sqrt{1-z^{-1}}}{\left(\sqrt{1-z^{% -1}}+1\right)^{2}}\right),$
$\|\operatorname{ph}\left(-z\right)\|<\pi$ , $\Re z<\frac{1}{2}$ .
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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, $z$ : complex variable, $a$ : real or complex parameter and $c$ : 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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, $z$ : complex variable, $a$ : real or complex parameter and $c$ : 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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, $z$ : complex variable, $a$ : real or complex parameter and $b$ : 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

$b=\tfrac{1}{3}a+\tfrac{1}{3}$ , $c=2b=a-b+1$ 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, $z$ : complex variable and $a$ : 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, $z$ : complex variable and $a$ : 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, $z$ : complex variable and $a$ : 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		$\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{3a}}\left(\frac{1}{\Gamma% \left(a+\frac{2}{3}\right)\Gamma\left(\frac{2}{3}\right)}F\left({a,a+\frac{1}{% 3}\atop\frac{2}{3}};z^{-3}\right)+\frac{{\mathrm{e}}^{\frac{1}{3}\pi\mathrm{i}% }}{z\Gamma\left(a\right)\Gamma\left(\frac{4}{3}\right)}F\left({a+\frac{1}{3},a% +\frac{2}{3}\atop\frac{4}{3}};z^{-3}\right)\right)=\frac{3^{\frac{3}{2}a+\frac% {1}{2}}{\mathrm{e}}^{\frac{1}{2}a\pi\mathrm{i}}\Gamma\left(a+\frac{1}{3}\right% )(1-\zeta)^{a}}{2\pi\Gamma\left(2a+\frac{2}{3}\right)(-\zeta)^{2a}}F\left({a+% \frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{-1}\right),$
$\|z\|>1$ , $\|\operatorname{ph}\left(-z\right)\|<\frac{1}{3}\pi$ .
ⓘ 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, $z$ : complex variable, $a$ : 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 $z$ : 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 $z$ 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 $\longrightarrow$ Group 3

Group 2 $\longrightarrow$ Group 3

Group 2 $\longrightarrow$ Group 1

Group 2 $\longrightarrow$ Group 4

Group 4 $\longrightarrow$ Group 2

§15.8(iv) Quadratic Transformations (Continued)

Examples

§15.8(v) Cubic Transformations

Examples

Ramanujan’s Cubic Transformation

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

§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

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

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