15 Hypergeometric Function Properties 15.9 Relations to Other Functions 15.11 Riemann’s Differential Equation

§15.10 Hypergeometric Differential Equation

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Referenced by:: §15.17(i)
Permalink:: http://dlmf.nist.gov/15.10
See also:: Annotations for Ch.15

§15.10(i) Fundamental Solutions
§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

§15.10(i) Fundamental Solutions

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Keywords:: fundamental solutions, hypergeometric differential equation, hypergeometric equation
A&S Ref:: 15.5
Notes:: See Andrews et al. (1999, §2.3) and Olver (1997b, pp. 163–168). (15.10.9) is a corrected version of (2.3.20) in the first reference.
Referenced by:: §13.2(i), Erratum (V1.0.1) for Clarifications
Permalink:: http://dlmf.nist.gov/15.10.i
Clarification (effective with 1.0.1):: Item (c) was clarified by stating more explicitly the roles of the parameters $a+n-1,b+n-1$ and $c$ .
See also:: Annotations for §15.10 and Ch.15

15.10.1		$z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii) Permalink: http://dlmf.nist.gov/15.10.E1 Encodings: TeX, pMML, png See also: Annotations for §15.10(i), §15.10 and Ch.15

This is the hypergeometric differential equation. It has regular singularities at $z=0,1,\infty$ , with corresponding exponent pairs $\{0,1-c\}$ , $\{0,c-a-b\}$ , $\{a,b\}$ , respectively. When none of the exponent pairs differ by an integer, that is, when none of $c$ , $c-a-b$ , $a-b$ is an integer, we have the following pairs $f_{1}(z)$ , $f_{2}(z)$ of fundamental solutions. They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity.

Singularity $z=0$

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Keywords:: Wronskians, hypergeometric differential equation, hypergeometric function, singularities
See also:: Annotations for §15.10(i), §15.10 and Ch.15

15.10.2	$\displaystyle f_{1}(z)$	$\displaystyle=F\left({a,b\atop c};z\right),$
15.10.2	$\displaystyle f_{2}(z)$	$\displaystyle=z^{1-c}F\left({a-c+1,b-c+1\atop 2-c};z\right),$
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions Referenced by: §15.10(i), §15.10(ii) Permalink: http://dlmf.nist.gov/15.10.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

15.10.3		$\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(1-c)z^{-c}(1-z)^{c-a-b-1}.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions Permalink: http://dlmf.nist.gov/15.10.E3 Encodings: TeX, pMML, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

Singularity $z=1$

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Keywords:: Wronskians, hypergeometric differential equation, hypergeometric function, singularities
See also:: Annotations for §15.10(i), §15.10 and Ch.15

15.10.4	$\displaystyle f_{1}(z)$	$\displaystyle=F\left({a,b\atop a+b+1-c};1-z\right),$
15.10.4	$\displaystyle f_{2}(z)$	$\displaystyle=(1-z)^{c-a-b}F\left({c-a,c-b\atop c-a-b+1};1-z\right),$
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions Referenced by: §15.10(ii) Permalink: http://dlmf.nist.gov/15.10.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

15.10.5		$\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(a+b-c)z^{-c}(1-z)^{c-a-b-1}.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions Permalink: http://dlmf.nist.gov/15.10.E5 Encodings: TeX, pMML, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

Singularity $z=\infty$

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Keywords:: Wronskians, fundamental solutions, hypergeometric differential equation, hypergeometric function, singularities
See also:: Annotations for §15.10(i), §15.10 and Ch.15

15.10.6	$\displaystyle f_{1}(z)$	$\displaystyle=z^{-a}F\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right),$
15.10.6	$\displaystyle f_{2}(z)$	$\displaystyle=z^{-b}F\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right),$
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions Referenced by: §15.10(ii) Permalink: http://dlmf.nist.gov/15.10.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

15.10.7		$\mathscr{W}\left\{f_{1}(z),f_{2}(z)\right\}=(a-b)z^{-c}(z-1)^{c-a-b-1}.$
ⓘ Symbols: $\mathscr{W}$ : Wronskian, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $f_{1}(z)$ , $f_{2}(z)$ : fundamental solutions Permalink: http://dlmf.nist.gov/15.10.E7 Encodings: TeX, pMML, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

(a) If $c$ equals $n=1,2,3,\dots$ , and $a=1,2,\dots,n-1$ , then fundamental solutions in the neighborhood of $z=0$ are given by (15.10.2) with the interpretation (15.2.5) for $f_{2}(z)$ .

(b) If $c$ equals $n=1,2,3,\dots$ , and $a\neq 1,2,\dots,n-1$ , then fundamental solutions in the neighborhood of $z=0$ are given by $F\left(a,b;n;z\right)$ and

15.10.8		$F\left({a,b\atop n};z\right)\ln z-\sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)!% {\left(1-a\right)_{k}}{\left(1-b\right)_{k}}}(-z)^{-k}\\ +\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{\left(b\right)_{k}}}{{\left(n% \right)_{k}}k!}z^{k}\left(\psi\left(a+k\right)+\psi\left(b+k\right)-\psi\left(% 1+k\right)-\psi\left(n+k\right)\right),$
$a,b\neq n-1,n-2,\dots,0,-1,-2,\dots$ ,
ⓘ 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), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $k$ : integer and $m$ : integer A&S Ref: 15.5.19. (Compared with (15.10.8) this result has a multiple of $F\left(a,b;n;z\right)$ added to the right-hand side. It also contains an error: the conditions on $a$ and $b$ should be $a,b\neq 1,2,\dots,m$ .) Referenced by: (15.10.8) Permalink: http://dlmf.nist.gov/15.10.E8 Encodings: TeX, pMML, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

15.10.9		$F\left({-m,b\atop n};z\right)\ln z-\sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-k-1)% !{\left(m+1\right)_{k}}{\left(1-b\right)_{k}}}(-z)^{-k}+\sum_{k=0}^{m}\frac{{% \left(-m\right)_{k}}{\left(b\right)_{k}}}{{\left(n\right)_{k}}k!}z^{k}\left(% \psi\left(1+m-k\right)+\psi\left(b+k\right)-\psi\left(1+k\right)-\psi\left(n+k% \right)\right)+(-1)^{m}m!\sum_{k=m+1}^{\infty}\frac{(k-1-m)!{\left(b\right)_{k% }}}{{\left(n\right)_{k}}k!}z^{k},$
$a=-m$ , $m=0,1,2,\dots$ ; $b\neq n-1,n-2,\dots,0,-1,-2,\dots$ ,
ⓘ 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), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $k$ : integer and $m$ : integer Referenced by: §15.10(i), §15.10(i) Permalink: http://dlmf.nist.gov/15.10.E9 Encodings: TeX, pMML, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

15.10.10		$F\left({-m,-\ell\atop n};z\right)\ln z-\sum_{k=1}^{n-1}\frac{(n-1)!(k-1)!}{(n-% k-1)!{\left(m+1\right)_{k}}{\left(\ell+1\right)_{k}}}(-z)^{-k}+\sum_{k=0}^{% \ell}\frac{{\left(-m\right)_{k}}{\left(-\ell\right)_{k}}}{{\left(n\right)_{k}}% k!}z^{k}\left(\psi\left(1+m-k\right)+\psi\left(1+\ell-k\right)-\psi\left(1+k% \right)-\psi\left(n+k\right)\right)+(-1)^{\ell}\ell\,!\sum_{k=\ell+1}^{m}\frac% {(k-1-\ell)!{\left(-m\right)_{k}}}{{\left(n\right)_{k}}k!}z^{k},$
$a=-m$ , $m=0,1,2,\dots$ ; $b=-\ell$ , $\ell=0,1,2,\dots,m$ .
ⓘ 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), $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $k$ : integer, $\ell$ : integer and $m$ : integer Referenced by: §15.10(i) Permalink: http://dlmf.nist.gov/15.10.E10 Encodings: TeX, pMML, png See also: Annotations for §15.10(i), §15.10(i), §15.10 and Ch.15

Moreover, in (15.10.9) and (15.10.10) the symbols $a$ and $b$ are interchangeable.

(c) If the parameter $c$ in the differential equation equals $2-n=0,-1,-2,\dots$ , then fundamental solutions in the neighborhood of $z=0$ are given by $z^{n-1}$ times those in (a) and (b), with $a$ and $b$ replaced throughout by $a+n-1$ and $b+n-1$ , respectively.

(d) If $a+b+1-c$ equals $n=1,2,3,\dots$ , or $2-n=0,-1,-2,\dots$ , then fundamental solutions in the neighborhood of $z=1$ are given by those in (a), (b), and (c) with $z$ replaced by $1-z$ .

(e) Finally, if $a-b+1$ equals $n=1,2,3,\dots$ , or $2-n=0,-1,-2,\dots$ , then fundamental solutions in the neighborhood of $z=\infty$ are given by $z^{-a}$ times those in (a), (b), and (c) with $b$ and $z$ replaced by $a-c+1$ and $\ifrac{1}{z}$ , respectively.

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

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Keywords:: Kummer’s solutions, hypergeometric differential equation
A&S Ref:: 15.5
Notes:: See Luke (1969a, Chapter III).
Referenced by:: §15.12(iii), §16.4(iii), §31.3(iii)
Permalink:: http://dlmf.nist.gov/15.10.ii
See also:: Annotations for §15.10 and Ch.15

The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions.

15.10.11	$\displaystyle w_{1}(z)$	$\displaystyle=F\left({a,b\atop c};z\right)=(1-z)^{c-a-b}F\left({c-a,c-b\atop c% };z\right)=(1-z)^{-a}F\left({a,c-b\atop c};\frac{z}{z-1}\right)=(1-z)^{-b}F% \left({c-a,b\atop c};\frac{z}{z-1}\right).$
ⓘ 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, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E11 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.12	$\displaystyle w_{2}(z)$	$\displaystyle={z^{1-c}}F\left({a-c+1,b-c+1\atop 2-c};z\right)$
		$\displaystyle={z^{1-c}(1-z)^{c-a-b}}\*F\left({1-a,1-b\atop 2-c};z\right)$
		$\displaystyle={z^{1-c}(1-z)^{c-a-1}}\*F\left({a-c+1,1-b\atop 2-c};\frac{z}{z-1% }\right)$
		$\displaystyle={z^{1-c}(1-z)^{c-b-1}}\*F\left({1-a,b-c+1\atop 2-c};\frac{z}{z-1% }\right).$
ⓘ 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, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E12 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.13	$\displaystyle w_{3}(z)$	$\displaystyle=F\left({a,b\atop a+b-c+1};1-z\right)$
		$\displaystyle=z^{1-c}F\left({a-c+1,b-c+1\atop a+b-c+1};1-z\right)$
		$\displaystyle=z^{-a}F\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)$
		$\displaystyle=z^{-b}F\left({b,b-c+1\atop a+b-c+1};1-\frac{1}{z}\right).$
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E13 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.14	$\displaystyle w_{4}(z)$	$\displaystyle=(1-z)^{c-a-b}F\left({c-a,c-b\atop c-a-b+1};1-z\right)$
		$\displaystyle=z^{1-c}(1-z)^{c-a-b}F\left({1-a,1-b\atop c-a-b+1};1-z\right)$
		$\displaystyle=z^{a-c}(1-z)^{c-a-b}F\left({1-a,c-a\atop c-a-b+1};1-\frac{1}{z}\right)$
		$\displaystyle=z^{b-c}(1-z)^{c-a-b}F\left({1-b,c-b\atop c-a-b+1};1-\frac{1}{z}% \right).$
ⓘ 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, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E14 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.15	$\displaystyle w_{5}(z)$	$\displaystyle=e^{a\pi\mathrm{i}}z^{-a}\*F\left({a,a-c+1\atop a-b+1};\frac{1}{z% }\right)$
		$\displaystyle=e^{(c-b)\pi\mathrm{i}}z^{b-c}(1-z)^{c-a-b}\*F\left({1-b,c-b\atop a% -b+1};\frac{1}{z}\right)$
		$\displaystyle=(1-z)^{-a}F\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)$
		$\displaystyle=e^{(c-1)\pi\mathrm{i}}z^{1-c}(1-z)^{c-a-1}\*F\left({1-b,a-c+1% \atop a-b+1};\frac{1}{1-z}\right).$
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E15 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.16	$\displaystyle w_{6}(z)$	$\displaystyle=e^{b\pi\mathrm{i}}z^{-b}F\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right)$
		$\displaystyle=e^{(c-a)\pi\mathrm{i}}z^{a-c}(1-z)^{c-a-b}\*F\left({1-a,c-a\atop b% -a+1};\frac{1}{z}\right)$
		$\displaystyle=(1-z)^{-b}F\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right)$
		$\displaystyle=e^{(c-1)\pi\mathrm{i}}z^{1-c}(1-z)^{c-b-1}\*F\left({1-a,b-c+1% \atop b-a+1};\frac{1}{1-z}\right).$
ⓘ 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, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E16 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15

The $\genfrac{(}{)}{0.0pt}{}{6}{3}=20$ connection formulas for the principal branches of Kummer’s solutions are:

15.10.17	$\displaystyle w_{3}(z)$	$\displaystyle=\frac{\Gamma\left(1-c\right)\Gamma\left(a+b-c+1\right)}{\Gamma% \left(a-c+1\right)\Gamma\left(b-c+1\right)}w_{1}(z)+\frac{\Gamma\left(c-1% \right)\Gamma\left(a+b-c+1\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_% {2}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E17 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.18	$\displaystyle w_{4}(z)$	$\displaystyle=\frac{\Gamma\left(1-c\right)\Gamma\left(c-a-b+1\right)}{\Gamma% \left(1-a\right)\Gamma\left(1-b\right)}w_{1}(z)+\frac{\Gamma\left(c-1\right)% \Gamma\left(c-a-b+1\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}w_{2}% (z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E18 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.19	$\displaystyle w_{5}(z)$	$\displaystyle=\frac{\Gamma\left(1-c\right)\Gamma\left(a-b+1\right)}{\Gamma% \left(a-c+1\right)\Gamma\left(1-b\right)}w_{1}(z)+e^{(c-1)\pi\mathrm{i}}\frac{% \Gamma\left(c-1\right)\Gamma\left(a-b+1\right)}{\Gamma\left(a\right)\Gamma% \left(c-b\right)}w_{2}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E19 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.20	$\displaystyle w_{6}(z)$	$\displaystyle=\frac{\Gamma\left(1-c\right)\Gamma\left(b-a+1\right)}{\Gamma% \left(b-c+1\right)\Gamma\left(1-a\right)}w_{1}(z)+e^{(c-1)\pi\mathrm{i}}\frac{% \Gamma\left(c-1\right)\Gamma\left(b-a+1\right)}{\Gamma\left(b\right)\Gamma% \left(c-a\right)}w_{2}(z).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E20 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15

15.10.21	$\displaystyle w_{1}(z)$	$\displaystyle=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{\Gamma\left(% c-a\right)\Gamma\left(c-b\right)}w_{3}(z)+\frac{\Gamma\left(c\right)\Gamma% \left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}w_{4}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Referenced by: §15.8(i) Permalink: http://dlmf.nist.gov/15.10.E21 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.22	$\displaystyle w_{2}(z)$	$\displaystyle=\frac{\Gamma\left(2-c\right)\Gamma\left(c-a-b\right)}{\Gamma% \left(1-a\right)\Gamma\left(1-b\right)}w_{3}(z)+\frac{\Gamma\left(2-c\right)% \Gamma\left(a+b-c\right)}{\Gamma\left(a-c+1\right)\Gamma\left(b-c+1\right)}w_{% 4}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E22 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.23	$\displaystyle w_{5}(z)$	$\displaystyle=e^{a\pi\mathrm{i}}\frac{\Gamma\left(a-b+1\right)\Gamma\left(c-a-% b\right)}{\Gamma\left(1-b\right)\Gamma\left(c-b\right)}w_{3}(z)+e^{(c-b)\pi% \mathrm{i}}\frac{\Gamma\left(a-b+1\right)\Gamma\left(a+b-c\right)}{\Gamma\left% (a\right)\Gamma\left(a-c+1\right)}w_{4}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E23 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.24	$\displaystyle w_{6}(z)$	$\displaystyle=e^{b\pi\mathrm{i}}\frac{\Gamma\left(b-a+1\right)\Gamma\left(c-a-% b\right)}{\Gamma\left(1-a\right)\Gamma\left(c-a\right)}w_{3}(z)+e^{(c-a)\pi% \mathrm{i}}\frac{\Gamma\left(b-a+1\right)\Gamma\left(a+b-c\right)}{\Gamma\left% (b\right)\Gamma\left(b-c+1\right)}w_{4}(z).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E24 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15

15.10.25	$\displaystyle w_{1}(z)$	$\displaystyle=\frac{\Gamma\left(c\right)\Gamma\left(b-a\right)}{\Gamma\left(b% \right)\Gamma\left(c-a\right)}w_{5}(z)+\frac{\Gamma\left(c\right)\Gamma\left(a% -b\right)}{\Gamma\left(a\right)\Gamma\left(c-b\right)}w_{6}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Referenced by: §15.8(i) Permalink: http://dlmf.nist.gov/15.10.E25 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.26	$\displaystyle w_{2}(z)$	$\displaystyle=e^{(1-c)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(b-% a\right)}{\Gamma\left(1-a\right)\Gamma\left(b-c+1\right)}w_{5}(z)+e^{(1-c)\pi% \mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(a-b\right)}{\Gamma\left(1-b% \right)\Gamma\left(a-c+1\right)}w_{6}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E26 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.27	$\displaystyle w_{3}(z)$	$\displaystyle=e^{-a\pi\mathrm{i}}\frac{\Gamma\left(a+b-c+1\right)\Gamma\left(b% -a\right)}{\Gamma\left(b\right)\Gamma\left(b-c+1\right)}w_{5}(z)+e^{-b\pi% \mathrm{i}}\frac{\Gamma\left(a+b-c+1\right)\Gamma\left(a-b\right)}{\Gamma\left% (a\right)\Gamma\left(a-c+1\right)}w_{6}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E27 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.28	$\displaystyle w_{4}(z)$	$\displaystyle=e^{(b-c)\pi\mathrm{i}}\frac{\Gamma\left(c-a-b+1\right)\Gamma% \left(b-a\right)}{\Gamma\left(1-a\right)\Gamma\left(c-a\right)}w_{5}(z)+e^{(a-% c)\pi\mathrm{i}}\frac{\Gamma\left(c-a-b+1\right)\Gamma\left(a-b\right)}{\Gamma% \left(1-b\right)\Gamma\left(c-b\right)}w_{6}(z).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E28 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15

15.10.29	$\displaystyle w_{1}(z)$	$\displaystyle=e^{b\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(a-c+1% \right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(c-b\right)}w_{3}(z)+e^{(b-c)\pi% \mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(a-c+1\right)}{\Gamma\left(b% \right)\Gamma\left(a-b+1\right)}w_{5}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E29 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.30	$\displaystyle w_{1}(z)$	$\displaystyle=e^{a\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(b-c+1% \right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(c-a\right)}w_{3}(z)+e^{(a-c)\pi% \mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(b-c+1\right)}{\Gamma\left(a% \right)\Gamma\left(b-a+1\right)}w_{6}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E30 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.31	$\displaystyle w_{2}(z)$	$\displaystyle=e^{(b-c+1)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(% a\right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(1-b\right)}w_{3}(z)+e^{(b-c)% \pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(a\right)}{\Gamma\left(a-% b+1\right)\Gamma\left(b-c+1\right)}w_{5}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E31 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.32	$\displaystyle w_{2}(z)$	$\displaystyle=e^{(a-c+1)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(% b\right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(1-a\right)}w_{3}(z)+e^{(a-c)% \pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(b\right)}{\Gamma\left(b-% a+1\right)\Gamma\left(a-c+1\right)}w_{6}(z).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E32 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15

15.10.33	$\displaystyle w_{1}(z)$	$\displaystyle=e^{(c-a)\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-b% \right)}{\Gamma\left(a\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-a\pi% \mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-b\right)}{\Gamma\left(a-b+1% \right)\Gamma\left(c-a\right)}w_{5}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E33 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.34	$\displaystyle w_{1}(z)$	$\displaystyle=e^{(c-b)\pi\mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-a% \right)}{\Gamma\left(b\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-b\pi% \mathrm{i}}\frac{\Gamma\left(c\right)\Gamma\left(1-a\right)}{\Gamma\left(b-a+1% \right)\Gamma\left(c-b\right)}w_{6}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E34 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.35	$\displaystyle w_{2}(z)$	$\displaystyle=e^{(1-a)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-% b\right)}{\Gamma\left(a-c+1\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-a\pi% \mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-b\right)}{\Gamma\left(a-b% +1\right)\Gamma\left(1-a\right)}w_{5}(z),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E35 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15
15.10.36	$\displaystyle w_{2}(z)$	$\displaystyle=e^{(1-b)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-% a\right)}{\Gamma\left(b-c+1\right)\Gamma\left(c-a-b+1\right)}w_{4}(z)+e^{-b\pi% \mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma\left(c-a\right)}{\Gamma\left(b-a% +1\right)\Gamma\left(1-b\right)}w_{6}(z).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $w_{m}(z)$ : Kummer’s solutions Permalink: http://dlmf.nist.gov/15.10.E36 Encodings: TeX, pMML, png See also: Annotations for §15.10(ii), §15.10 and Ch.15

© 2010–2020 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.1.0; Release date 2020-12-15. A printed companion is available. 15.9 Relations to Other Functions 15.11 Riemann’s Differential Equation

§15.10 Hypergeometric Differential Equation

Contents

§15.10(i) Fundamental Solutions

Singularity z=0

Singularity z=1

Singularity z=∞

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

Singularity $z=0$

Singularity $z=1$

Singularity $z=\infty$