§15.10 Hypergeometric Differential Equation

Referenced by:: §15.17(i)
Permalink:: http://dlmf.nist.gov/15.10

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

§15.10(i) Fundamental Solutions

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.
A&S Ref:: 15.5
Keywords:: hypergeometric differential equation, hypergeometric equation
Referenced by:: ¶ ‣ §13.2(i), Version 1.0.1 (June 27, 2011)
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 $c$ .
Reported 2011-02-15

15.10.1 $z(1-z)\frac{{d}^{2}w}{{dz}^{2}}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$

Symbols:: $\frac{df}{dx}$ : 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
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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 $z=0$

Keywords:: hypergeometric differential equation, hypergeometric function, singularities

15.10.2

$f_{1}(z)=\mathop{F\/}\nolimits\!\left({a,b\atop c};z\right),$

$f_{2}(z)=z^{{1-c}}\mathop{F\/}\nolimits\!\left({a-c+1,b-c+1\atop 2-c};z\right),$

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

15.10.3 $\mathop{\mathscr{W}\/}\nolimits\left\{ f_{1}(z),f_{2}(z)\right\}=(1-c)z^{{-c}}(1-z)^{{c-a-b-1}}.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : 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

¶ Singularity $z=1$

Keywords:: hypergeometric differential equation, hypergeometric function, singularities

15.10.4

$f_{1}(z)=\mathop{F\/}\nolimits\!\left({a,b\atop a+b+1-c};1-z\right),$

$f_{2}(z)=(1-z)^{{c-a-b}}\mathop{F\/}\nolimits\!\left({c-a,c-b\atop c-a-b+1};1-z\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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
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15.10.5 $\mathop{\mathscr{W}\/}\nolimits\left\{ f_{1}(z),f_{2}(z)\right\}=(a+b-c)z^{{-c}}(1-z)^{{c-a-b-1}}.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : 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
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¶ Singularity $z=\infty$

Keywords:: hypergeometric differential equation, hypergeometric function, singularities

15.10.6

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

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

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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
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15.10.7 $\mathop{\mathscr{W}\/}\nolimits\left\{ f_{1}(z),f_{2}(z)\right\}=(a-b)z^{{-c}}(z-1)^{{c-a-b-1}}.$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : 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

(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 $\mathop{F\/}\nolimits\!\left(a,b;n;z\right)$ and

15.10.8 $\mathop{F\/}\nolimits\!\left({a,b\atop n};z\right)\mathop{\ln\/}\nolimits z-\sum _{{k=1}}^{{n-1}}\frac{(n-1)!(k-1)!}{(n-k-1)!\left(1-a\right)_{{k}}\left(1-b\right)_{{k}}}(-z)^{{-k}}\\ +\sum _{{k=0}}^{{\infty}}\frac{\left(a\right)_{{k}}\left(b\right)_{{k}}}{\left(n\right)_{{k}}k!}z^{k}\left(\mathop{\psi\/}\nolimits\!\left(a+k\right)+\mathop{\psi\/}\nolimits\!\left(b+k\right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!\left(n+k\right)\right),$ $a,b\neq n-1,n-2,\dots,0,-1,-2,\dots$ ,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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 $\mathop{F\/}\nolimits\!\left(a,b;n;z\right)$ 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

15.10.9 $\mathop{F\/}\nolimits\!\left({-m,b\atop n};z\right)\mathop{\ln\/}\nolimits z-\sum _{{k=1}}^{{n-1}}\frac{(n-1)!(k-1)!}{(n-k-1)!\left(m+1\right)_{{k}}\left(1-b\right)_{{k}}}(-z)^{{-k}}+\sum _{{k=0}}^{m}\frac{\left(-m\right)_{{k}}\left(b\right)_{{k}}}{\left(n\right)_{{k}}k!}z^{k}\left(\mathop{\psi\/}\nolimits\!\left(1+m-k\right)+\mathop{\psi\/}\nolimits\!\left(b+k\right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!\left(n+k\right)\right)+(-1)^{m}m!\sum _{{k=m+1}}^{{\infty}}\frac{(k-1-m)!\left(b\right)_{{k}}}{\left(n\right)_{{k}}k!}z^{k},$ $a=-m$ , $m=0,1,2,\dots$ ; $b\neq n-1,n-2,\dots,0,-1,-2,\dots$ ,

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

15.10.10 $\mathop{F\/}\nolimits\!\left({-m,-\ell\atop n};z\right)\mathop{\ln\/}\nolimits z-\sum _{{k=1}}^{{n-1}}\frac{(n-1)!(k-1)!}{(n-k-1)!\left(m+1\right)_{{k}}\left(\ell+1\right)_{{k}}}(-z)^{{-k}}+\sum _{{k=0}}^{{\ell}}\frac{\left(-m\right)_{{k}}\left(-\ell\right)_{{k}}}{\left(n\right)_{{k}}k!}z^{k}\left(\mathop{\psi\/}\nolimits\!\left(1+m-k\right)+\mathop{\psi\/}\nolimits\!\left(1+\ell-k\right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop{\psi\/}\nolimits\!\left(n+k\right)\right)+(-1)^{\ell}\ell\,!\sum _{{k=\ell+1}}^{m}\frac{(k-1-\ell)!\left(-m\right)_{{k}}}{\left(n\right)_{{k}}k!}z^{k},$ $a=-m$ , $m=0,1,2,\dots$ ; $b=-\ell$ , $\ell=0,1,2,\dots,m$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : 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

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

Notes:: See Luke (1969a, Chapter III).
A&S Ref:: 15.5
Keywords:: hypergeometric differential equation
Referenced by:: §15.12(iii), §16.4(iii), §31.3(iii)
Permalink:: http://dlmf.nist.gov/15.10.ii

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

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

15.10.12 $w_{2}(z)={z^{{1-c}}}\mathop{F\/}\nolimits\!\left({a-c+1,b-c+1\atop 2-c};z\right)={z^{{1-c}}(1-z)^{{c-a-b}}}\*\mathop{F\/}\nolimits\!\left({1-a,1-b\atop 2-c};z\right)={z^{{1-c}}(1-z)^{{c-a-1}}}\*\mathop{F\/}\nolimits\!\left({a-c+1,1-b\atop 2-c};\frac{z}{z-1}\right)={z^{{1-c}}(1-z)^{{c-b-1}}}\*\mathop{F\/}\nolimits\!\left({1-a,b-c+1\atop 2-c};\frac{z}{z-1}\right).$

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

15.10.13 $w_{3}(z)=\mathop{F\/}\nolimits\!\left({a,b\atop a+b-c+1};1-z\right)=z^{{1-c}}\mathop{F\/}\nolimits\!\left({a-c+1,b-c+1\atop a+b-c+1};1-z\right)=z^{{-a}}\mathop{F\/}\nolimits\!\left({a,a-c+1\atop a+b-c+1};1-\frac{1}{z}\right)=z^{{-b}}\mathop{F\/}\nolimits\!\left({b,b-c+1\atop a+b-c+1};1-\frac{1}{z}\right).$

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

15.10.14 $w_{4}(z)=(1-z)^{{c-a-b}}\mathop{F\/}\nolimits\!\left({c-a,c-b\atop c-a-b+1};1-z\right)=z^{{1-c}}(1-z)^{{c-a-b}}\mathop{F\/}\nolimits\!\left({1-a,1-b\atop c-a-b+1};1-z\right)=z^{{a-c}}(1-z)^{{c-a-b}}\mathop{F\/}\nolimits\!\left({1-a,c-a\atop c-a-b+1};1-\frac{1}{z}\right)=z^{{b-c}}(1-z)^{{c-a-b}}\mathop{F\/}\nolimits\!\left({1-b,c-b\atop c-a-b+1};1-\frac{1}{z}\right).$

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

15.10.15 $w_{5}(z)=e^{{a\pi i}}z^{{-a}}\*\mathop{F\/}\nolimits\!\left({a,a-c+1\atop a-b+1};\frac{1}{z}\right)=e^{{(c-b)\pi i}}z^{{b-c}}(1-z)^{{c-a-b}}\*\mathop{F\/}\nolimits\!\left({1-b,c-b\atop a-b+1};\frac{1}{z}\right)=(1-z)^{{-a}}\mathop{F\/}\nolimits\!\left({a,c-b\atop a-b+1};\frac{1}{1-z}\right)=e^{{(c-1)\pi i}}z^{{1-c}}(1-z)^{{c-a-1}}\*\mathop{F\/}\nolimits\!\left({1-b,a-c+1\atop a-b+1};\frac{1}{1-z}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $e$ : base of exponential 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.E15
Encodings:: TeX, pMML, png

15.10.16 $w_{6}(z)=e^{{b\pi i}}z^{{-b}}\mathop{F\/}\nolimits\!\left({b,b-c+1\atop b-a+1};\frac{1}{z}\right)=e^{{(c-a)\pi i}}z^{{a-c}}(1-z)^{{c-a-b}}\*\mathop{F\/}\nolimits\!\left({1-a,c-a\atop b-a+1};\frac{1}{z}\right)=(1-z)^{{-b}}\mathop{F\/}\nolimits\!\left({b,c-a\atop b-a+1};\frac{1}{1-z}\right)=e^{{(c-1)\pi i}}z^{{1-c}}(1-z)^{{c-b-1}}\*\mathop{F\/}\nolimits\!\left({1-a,b-c+1\atop b-a+1};\frac{1}{1-z}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $e$ : base of exponential 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.E16
Encodings:: TeX, pMML, png

The $\binom{6}{3}=20$ connection formulas for the principal branches of Kummer’s solutions are:

15.10.17 $w_{3}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{1}(z)+\frac{\mathop{\Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}w_{2}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

15.10.18 $w_{4}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{1}(z)+\frac{\mathop{\Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{2}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

15.10.19 $w_{5}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{1}(z)+e^{{(c-1)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{2}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E19
Encodings:: TeX, pMML, png

15.10.20 $w_{6}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1-c\right)\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}w_{1}(z)+e^{{(c-1)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c-1\right)\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{2}(z).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E20
Encodings:: TeX, pMML, png

15.10.21 $w_{1}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{3}(z)+\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}w_{4}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

15.10.22 $w_{2}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{3}(z)+\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{4}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

15.10.23 $w_{5}(z)=e^{{a\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{3}(z)+e^{{(c-b)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{4}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E23
Encodings:: TeX, pMML, png

15.10.24 $w_{6}(z)=e^{{b\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{3}(z)+e^{{(c-a)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{4}(z).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E24
Encodings:: TeX, pMML, png

15.10.25 $w_{1}(z)=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{5}(z)+\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{6}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : 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

15.10.26 $w_{2}(z)=e^{{(1-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{5}(z)+e^{{(1-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{6}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E26
Encodings:: TeX, pMML, png

15.10.27 $w_{3}(z)=e^{{-a\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{5}(z)+e^{{-b\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{6}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E27
Encodings:: TeX, pMML, png

15.10.28 $w_{4}(z)=e^{{(b-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{5}(z)+e^{{(a-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(1-b\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{6}(z).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E28
Encodings:: TeX, pMML, png

15.10.29 $w_{1}(z)=e^{{b\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{3}(z)+e^{{(b-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}w_{5}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E29
Encodings:: TeX, pMML, png

15.10.30 $w_{1}(z)=e^{{a\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{3}(z)+e^{{(a-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)}w_{6}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E30
Encodings:: TeX, pMML, png

15.10.31 $w_{2}(z)=e^{{(b-c+1)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{3}(z)+e^{{(b-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(a\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)}w_{5}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E31
Encodings:: TeX, pMML, png

15.10.32 $w_{2}(z)=e^{{(a-c+1)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}w_{3}(z)+e^{{(a-c)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)}w_{6}(z).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E32
Encodings:: TeX, pMML, png

15.10.33 $w_{1}(z)=e^{{(c-a)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^{{-a\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}w_{5}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E33
Encodings:: TeX, pMML, png

15.10.34 $w_{1}(z)=e^{{(c-b)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^{{-b\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}w_{6}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E34
Encodings:: TeX, pMML, png

15.10.35 $w_{2}(z)=e^{{(1-a)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^{{-a\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}w_{5}(z),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E35
Encodings:: TeX, pMML, png

15.10.36 $w_{2}(z)=e^{{(1-b)\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-c+1\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b+1\right)}w_{4}(z)+e^{{-b\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(2-c\right)\mathop{\Gamma\/}\nolimits\!\left(c-a\right)}{\mathop{\Gamma\/}\nolimits\!\left(b-a+1\right)\mathop{\Gamma\/}\nolimits\!\left(1-b\right)}w_{6}(z).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $e$ : base of exponential 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.E36
Encodings:: TeX, pMML, png

§15.10 Hypergeometric Differential Equation

Contents

§15.10(i) Fundamental Solutions

¶ Singularity

¶ Singularity

¶ Singularity

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

¶ Singularity $z=0$

¶ Singularity $z=1$

¶ Singularity $z=\infty$