§15.8 Transformations of Variable

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

15.8.1		$$\begin{array}{ll}\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)& ={(1-z)}^{-a}\mathbf{F}(\genfrac{}{}{0pt}{}{a,c-b}{c};\frac{z}{z-1})={(1-z)}^{-b}\mathbf{F}(\genfrac{}{}{0pt}{}{c-a,b}{c};\frac{z}{z-1})\\ & ={(1-z)}^{c-a-b}\mathbf{F}(\genfrac{}{}{0pt}{}{c-a,c-b}{c};z),\end{array}$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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(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		${\displaystyle \frac{\sin \left(\pi (b-a)\right)}{\pi }}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,b}{c}};z)$	$={\displaystyle \frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,a-c+1}{a-b+1}};{\displaystyle \frac{1}{z}})-{\displaystyle \frac{{(-z)}^{-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{b,b-c+1}{b-a+1}};{\displaystyle \frac{1}{z}}),$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: 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		${\displaystyle \frac{\sin \left(\pi (b-a)\right)}{\pi }}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,b}{c}};z)$	$={\displaystyle \frac{{(1-z)}^{-a}}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,c-b}{a-b+1}};{\displaystyle \frac{1}{1-z}})-{\displaystyle \frac{{(1-z)}^{-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{b,c-a}{b-a+1}};{\displaystyle \frac{1}{1-z}}),$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: 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		${\displaystyle \frac{\sin \left(\pi (c-a-b)\right)}{\pi }}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,b}{c}};z)$	$={\displaystyle \frac{1}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,b}{a+b-c+1}};1-z)-{\displaystyle \frac{{(1-z)}^{c-a-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{c-a,c-b}{c-a-b+1}};1-z),$
, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: 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		${\displaystyle \frac{\sin \left(\pi (c-a-b)\right)}{\pi }}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,b}{c}};z)$	$={\displaystyle \frac{z^{-a}}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{a,a-c+1}{a+b-c+1}};1-{\displaystyle \frac{1}{z}})-{\displaystyle \frac{{(1-z)}^{c-a-b}{z}^{a-c}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}}\mathbf{F}({\displaystyle \genfrac{}{}{0pt}{}{c-a,1-a}{c-a-b+1}};1-{\displaystyle \frac{1}{z}}),$
, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $\sin z$: 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).

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

15.8.6		$F({\displaystyle \genfrac{}{}{0pt}{}{-m,b}{c}};z)$	$={\displaystyle \frac{{(b)}_m}{{(c)}_m}}{(-z)}^{m}F({\displaystyle \genfrac{}{}{0pt}{}{-m,1-c-m}{1-b-m}};{\displaystyle \frac{1}{z}})={\displaystyle \frac{{(b)}_m}{{(c)}_m}}{(1-z)}^{m}F({\displaystyle \genfrac{}{}{0pt}{}{-m,c-b}{1-b-m}};{\displaystyle \frac{1}{1-z}}),$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ 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 See also: Annotations for §15.8(ii), §15.8 and Ch.15
15.8.7		$F({\displaystyle \genfrac{}{}{0pt}{}{-m,b}{c}};z)$	$={\displaystyle \frac{{(c-b)}_m}{{(c)}_m}}F({\displaystyle \genfrac{}{}{0pt}{}{-m,b}{b-c-m+1}};1-z)={\displaystyle \frac{{(c-b)}_m}{{(c)}_m}}{z}^{m}F({\displaystyle \genfrac{}{}{0pt}{}{-m,1-c-m}{b-c-m+1}};1-{\displaystyle \frac{1}{z}}),$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ 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 See also: Annotations for §15.8(ii), §15.8 and Ch.15

with the understanding that if $b=-\mathrm{\ell }$, $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, then $m\le \mathrm{\ell }$.

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		$$\mathbf{F}(\genfrac{}{}{0pt}{}{a,a+m}{c};z)=\begin{array}{l}\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(a+m\right)}\sum _{k=0}^{m-1}\frac{{(a)}_k(m-k-1)!}{k!\mathrm{\Gamma }\left(c-a-k\right)}{z}^{-k}\\ +\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\begin{array}{l}\frac{{(a+m)}_k}{k!(k+m)!\mathrm{\Gamma }\left(c-a-k-m\right)}{(-1)}^k{z}^{-k-m}\\ \times (\begin{array}{l}\ln \left(-z\right)+\psi \left(k+1\right)+\psi \left(k+m+1\right)\\ -\psi \left(a+k+m\right)-\psi \left(c-a-k-m\right)),\end{array}\end{array}\end{array}$$
,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 See also: Annotations for §15.8(ii), §15.8 and Ch.15

15.8.9		$$\mathbf{F}(\genfrac{}{}{0pt}{}{a,a+m}{c};z)=\begin{array}{l}\frac{{(1-z)}^{-a}}{\mathrm{\Gamma }\left(a+m\right)\mathrm{\Gamma }\left(c-a\right)}\sum _{k=0}^{m-1}\frac{{(a)}_k{(c-a-m)}_k(m-k-1)!}{k!}{(z-1)}^{-k}\\ +\frac{{(-1)}^m{(1-z)}^{-a-m}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-a-m\right)}\sum _{k=0}^{\mathrm{\infty }}\begin{array}{l}\frac{{(a+m)}_k{(c-a)}_k}{k!(k+m)!}{(1-z)}^{-k}\\ \times (\begin{array}{l}\ln \left(1-z\right)+\psi \left(k+1\right)+\psi \left(k+m+1\right)\\ -\psi \left(a+k+m\right)-\psi \left(c-a+k\right)),\end{array}\end{array}\end{array}$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 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 $\psi \left(c-a-k-m\right)/\mathrm{\Gamma }\left(c-a-k-m\right)$ 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 $\mathbf{F}(a,b;c;z)$.

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		$$\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{a+b+m};z)=\begin{array}{l}\frac{1}{\mathrm{\Gamma }\left(a+m\right)\mathrm{\Gamma }\left(b+m\right)}\sum _{k=0}^{m-1}\frac{{(a)}_k{(b)}_k(m-k-1)!}{k!}{(z-1)}^k\\ -\frac{{(z-1)}^m}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{k=0}^{\mathrm{\infty }}\begin{array}{l}\frac{{(a+m)}_k{(b+m)}_k}{k!(k+m)!}{(1-z)}^k\\ \times (\begin{array}{l}\ln \left(1-z\right)-\psi \left(k+1\right)-\psi \left(k+m+1\right)\\ +\psi \left(a+k+m\right)+\psi \left(b+k+m\right)),\end{array}\end{array}\end{array}$$
,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 See also: Annotations for §15.8(ii), §15.8 and Ch.15

15.8.11		$$\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{a+b+m};z)=\begin{array}{l}\frac{z^{-a}}{\mathrm{\Gamma }\left(a+m\right)}\sum _{k=0}^{m-1}\frac{{(a)}_k(m-k-1)!}{k!\mathrm{\Gamma }\left(b+m-k\right)}{\left(1-\frac{1}{z}\right)}^k\\ -\frac{z^{-a}}{\mathrm{\Gamma }\left(a\right)}\sum _{k=0}^{\mathrm{\infty }}\begin{array}{l}\frac{{(a+m)}_k}{k!(k+m)!\mathrm{\Gamma }\left(b-k\right)}{(-1)}^k{\left(1-\frac{1}{z}\right)}^{k+m}\\ \times (\begin{array}{l}\ln \left(\frac{1-z}{z}\right)-\psi \left(k+1\right)-\psi \left(k+m+1\right)\\ +\psi \left(a+k+m\right)+\psi \left(b-k\right)),\end{array}\end{array}\end{array}$$
.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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 See also: Annotations for §15.8(ii), §15.8 and Ch.15

In (15.8.11) when $b-k$ is a nonpositive integer, $\psi \left(b-k\right)/\mathrm{\Gamma }\left(b-k\right)$ 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		$$\mathbf{F}(a,b;a+b-m;z)={(1-z)}^{-m}\mathbf{F}(\tilde{a},\tilde{b};\tilde{a}+\tilde{b}+m;z),$$
$\tilde{a}=a-m,\tilde{b}=b-m$.
ⓘ Symbols: $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ 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

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.

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.