§15.8 Transformations of Variable

Permalink:: http://dlmf.nist.gov/15.8

§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

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).
A&S Ref:: 15.3
Keywords:: hypergeometric function
Referenced by:: §15.12(iii), §15.19(i), Version 1.0.1 (June 27, 2011)
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).
Reported 2011-03-02

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

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

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $\mathop{\sin\/}\nolimits z$ : 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

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, $\mathop{\sin\/}\nolimits z$ : 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

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

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).
A&S Ref:: 15.3
Keywords:: hypergeometric function
Referenced by:: §15.8(i)
Permalink:: http://dlmf.nist.gov/15.8.ii

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

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $z$ : 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

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $z$ : 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)
Permalink:: http://dlmf.nist.gov/15.8.E7
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

In (15.8.8) when $c-a-k-m$ is a nonpositive integer $\ifrac{\mathop{\psi\/}\nolimits\!\left(c-a-k-m\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a-k-m\right)}$ 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 $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ .

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

In (15.8.11) when $b-k$ is a nonpositive integer, $\ifrac{\mathop{\psi\/}\nolimits\!\left(b-k\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-k\right)}$ 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 $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;a+b-m;z\right)=(1-z)^{{-m}}\mathop{\mathbf{F}\/}\nolimits\!\left(\tilde{a},\tilde{b};\tilde{a}+\tilde{b}+m;z\right),$ $\tilde{a}=a-m,\tilde{b}=b-m$ .

Symbols:: $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : 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

§15.8(iii) Quadratic Transformations

Notes:: See Andrews et al. (1999, §3.1).
A&S Ref:: 15.3
Keywords:: hypergeometric function
Referenced by:: §14.3(iii), §15.17(i), §15.9(iv), §31.7(i)
Permalink:: http://dlmf.nist.gov/15.8.iii

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

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

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

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

¶ Group 2 $\longrightarrow$ Group 3

15.8.15 $\mathop{F\/}\nolimits\!\left({a,b\atop a-b+1};z\right)=(1+z)^{{-a}}\mathop{F\/}\nolimits\!\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop a-b+1};\frac{4z}{(1+z)^{2}}\right),$ $|z|<1$ ,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.8.16 $\mathop{F\/}\nolimits\!\left({a,b\atop a-b+1};z\right)=(1-z)^{{-a}}\mathop{F\/}\nolimits\!\left({\frac{1}{2}a,\frac{1}{2}a-b+\frac{1}{2}\atop a-b+1};\frac{-4z}{(1-z)^{2}}\right),$ $|z|<1$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.8.17 $\mathop{F\/}\nolimits\!\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=(1-2z)^{{-a}}\mathop{F\/}\nolimits\!\left({\frac{1}{2}a,\frac{1}{2}a+\frac{1}{2}\atop\frac{1}{2}(a+b+1)};\frac{4z(z-1)}{(1-2z)^{2}}\right),$ $\realpart{z}<\frac{1}{2}$ ,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\realpart{}$ : 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

15.8.18 $\mathop{F\/}\nolimits\!\left({a,b\atop\frac{1}{2}(a+b+1)};z\right)=\mathop{F\/}\nolimits\!\left({\frac{1}{2}a,\frac{1}{2}b\atop\frac{1}{2}(a+b+1)};4z(1-z)\right),$ $\realpart{z}<\frac{1}{2}$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\realpart{}$ : 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

15.8.19 $\mathop{F\/}\nolimits\!\left({a,1-a\atop c};z\right)=(1-2z)^{{1-a-c}}(1-z)^{{c-1}}\mathop{F\/}\nolimits\!\left({\frac{1}{2}(a+c),\frac{1}{2}(a+c-1)\atop c};\frac{4z(z-1)}{(1-2z)^{2}}\right),$ $\realpart{z}<\frac{1}{2}$ ,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\realpart{}$ : 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

15.8.20 $\mathop{F\/}\nolimits\!\left({a,1-a\atop c};z\right)=(1-z)^{{c-1}}\mathop{F\/}\nolimits\!\left({\frac{1}{2}(c-a),\frac{1}{2}(a+c-1)\atop c};4z(1-z)\right),$ $\realpart{z}<\frac{1}{2}$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\realpart{}$ : 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

¶ Group 2 $\longrightarrow$ Group 1

15.8.21 $\mathop{F\/}\nolimits\!\left({a,b\atop a-b+1};z\right)=\left(1+\sqrt{z}\right)^{{-2a}}\mathop{F\/}\nolimits\!\left({a,a-b+\tfrac{1}{2}\atop 2a-2b+1};\frac{4\sqrt{z}}{(1+\sqrt{z})^{2}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ , $|z|<1$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

15.8.22 $\mathop{F\/}\nolimits\!\left({a,b\atop\tfrac{1}{2}(a+b+1)};z\right)=\left(\frac{\sqrt{1-z^{{-1}}}-1}{\sqrt{1-z^{{-1}}}+1}\right)^{a}\mathop{F\/}\nolimits\!\left({a,\tfrac{1}{2}(a+b)\atop a+b};\frac{4\sqrt{1-z^{{-1}}}}{\left(\sqrt{1-z^{{-1}}}+1\right)^{2}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)|<\pi$ , $\realpart{z}<\frac{1}{2}$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : 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

15.8.23 $\mathop{F\/}\nolimits\!\left({a,1-a\atop c};z\right)=\left(\sqrt{1-z^{{-1}}}-1\right)^{{1-a}}\left(\sqrt{1-z^{{-1}}}+1\right)^{{a-2c+1}}\left(1-z^{{-1}}\right)^{{c-1}}\mathop{F\/}\nolimits\!\left({c-a,c-\tfrac{1}{2}\atop 2c-1};\frac{4\sqrt{1-z^{{-1}}}}{\left(\sqrt{1-z^{{-1}}}+1\right)^{2}}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)|<\pi$ , $\realpart{z}<\frac{1}{2}$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : 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

¶ Group 2 $\longrightarrow$ Group 4

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

¶ Group 4 $\longrightarrow$ Group 2

Keywords:: hypergeometric function

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

§15.8(iv) Quadratic Transformations (Continued)

Permalink:: http://dlmf.nist.gov/15.8.iv

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

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

(15.8.21) becomes

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.8.E29
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

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

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).
Keywords:: hypergeometric function
Referenced by:: §31.7(i)
Permalink:: http://dlmf.nist.gov/15.8.v

¶ Examples

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\realpart{}$ : 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

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

15.8.32 $\frac{\left(1-z^{3}\right)^{a}}{\left(-z\right)^{{3a}}}\left(\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a+\frac{2}{3}\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{2}{3}\right)}\mathop{F\/}\nolimits\!\left({a,a+\frac{1}{3}\atop\frac{2}{3}};z^{{-3}}\right)+\frac{e^{{\frac{1}{3}\pi i}}}{z\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(\frac{4}{3}\right)}\mathop{F\/}\nolimits\!\left({a+\frac{1}{3},a+\frac{2}{3}\atop\frac{4}{3}};z^{{-3}}\right)\right)=\frac{3^{{\frac{3}{2}a+\frac{1}{2}}}e^{{\frac{1}{2}a\pi i}}\mathop{\Gamma\/}\nolimits\!\left(a+\frac{1}{3}\right)(1-\zeta)^{a}}{2\pi\mathop{\Gamma\/}\nolimits\!\left(2a+\frac{2}{3}\right)(-\zeta)^{{2a}}}\mathop{F\/}\nolimits\!\left({a+\frac{1}{3},3a\atop 2a+\frac{2}{3}};\zeta^{{-1}}\right),$ $|z|>1$ , $|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)|<\frac{1}{3}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $e$ : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

¶ Ramanujan’s Cubic Transformation

Keywords:: Ramanujan’s cubic transformation, hypergeometric function

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function and $z$ : complex variable
Referenced by:: §15.8(v)
Permalink:: http://dlmf.nist.gov/15.8.E33
Encodings:: TeX, pMML, png

provided that $z$ lies in the intersection of the open disks $\left|z-\frac{1}{4}\pm\frac{1}{4}\sqrt{3}i\right|<\frac{1}{2}\sqrt{3}$ , or equivalently, $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(\ifrac{(1-z)}{(1+2z)}\right)\right|<\pi/3$ . 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), and Vidūnas (2005); see also Erdélyi et al. (1953a, pp. 67 and 113–114).

§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