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An ordinary second-order Fuchsian linear differential equation having precisely three singular points:
 
An ordinary second-order Fuchsian linear differential equation having precisely three singular points:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
w ^ {\prime\prime } + \left (
 +
\frac{1 - \alpha - \alpha  ^  \prime  }{z-
 +
a} +
 +
\frac{1 - \beta - \beta  ^  \prime
 +
}{z-
 +
b} +
 +
\frac{1- \gamma - \gamma  ^  \prime  }{z-
 +
c} \right ) w  ^  \prime  +
 +
$$
 +
 
 +
$$
 +
+
 +
\left [
 +
\frac{\alpha \alpha  ^  \prime  ( a- b)( a- c) }{z-
 +
a}
 +
+
 +
\frac{\beta \beta  ^  \prime  ( b- c)( b- a) }{z-
 +
b}\right . +
 +
$$
 +
 
 +
$$
 +
+ \left .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711302.png" /></td> </tr></table>
+
\frac{\gamma \gamma  ^  \prime  ( c- a)( c- b) }{z-
 +
c} \right ]
 +
\frac{w}{(z- a)( z- b)( z- c)}  = 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711303.png" /></td> </tr></table>
+
$$
 +
\alpha + \alpha  ^  \prime  + \beta + \beta  ^  \prime  + \gamma + \gamma  ^  \prime  = 1;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711304.png" /></td> </tr></table>
+
here  $  a, b, c $
 +
are pairwise distinct complex numbers,  $  \alpha , \alpha  ^  \prime  $(
 +
$  \beta , \beta  ^  \prime  $
 +
and  $  \gamma , \gamma  ^  \prime  $)
 +
are the characteristic exponents at the singular point  $  z= a $(
 +
respectively,  $  z= b $
 +
and  $  z= c $).  
 +
A Papperitz equation is uniquely determined by the assignment of the singular points and the characteristic exponents. In solving a Papperitz equation (1), use is made of Riemann's notation:
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711305.png" /> are pairwise distinct complex numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711306.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711308.png" />) are the characteristic exponents at the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p0711309.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113011.png" />). A Papperitz equation is uniquely determined by the assignment of the singular points and the characteristic exponents. In solving a Papperitz equation (1), use is made of Riemann's notation:
+
$$
 +
= P \left \{
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113012.png" /></td> </tr></table>
+
\begin{array}{llll}
 +
a  & b  & c  &{}  \\
 +
\alpha  &\beta  &\gamma  & z  \\
 +
\alpha  ^  \prime  &\beta  ^  \prime  &\gamma  ^  \prime  &{}  \\
 +
\end{array}
 +
\right \} .
 +
$$
  
B. Riemann investigated [[#References|[1]]] the problem of finding all many-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113013.png" />, analytic in the extended complex plane, which have the following properties:
+
B. Riemann investigated [[#References|[1]]] the problem of finding all many-valued functions $  w( z) $,  
 +
analytic in the extended complex plane, which have the following properties:
  
a) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113014.png" /> has precisely three singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113015.png" />;
+
a) the function $  w( z) $
 +
has precisely three singular points $  a, b, c $;
  
 
b) any three of its branches are connected by a linear equation
 
b) any three of its branches are connected by a linear equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113016.png" /></td> </tr></table>
+
$$
 +
A _ {1} w _ {1} ( z) + A _ {2} w _ {2} ( z) + A _ {3} w _ {3} ( z)  = 0
 +
$$
  
 
with constant coefficients;
 
with constant coefficients;
  
c) the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113017.png" /> has the simplest singularities at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113018.png" />; namely, in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113019.png" /> there are two branches <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113021.png" /> satisfying
+
c) the function $  w( z) $
 +
has the simplest singularities at the points $  a, b, c $;  
 +
namely, in a neighbourhood of the point $  z= a $
 +
there are two branches $  \widetilde{w}  _ {1} ( z) $
 +
and $  \widetilde{w}  _ {2} ( z) $
 +
satisfying
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113022.png" /></td> </tr></table>
+
$$
 +
\widetilde{w}  _ {1} ( z)  = \
 +
( z- a)  ^  \alpha  \phi _ {1} ( z) ,\ \
 +
\widetilde{w}  _ {2} ( z)  = \
 +
( z- a) ^ {\alpha  ^  \prime  } \phi _ {2} ( z) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113023.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113024.png" /> is holomorphic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113025.png" />; and analogously for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113027.png" />.
+
where $  \phi _ {j} ( z) $
 +
$  ( j = 1, 2) $
 +
is holomorphic at $  z= a $;  
 +
and analogously for $  b $
 +
and $  c $.
  
Riemann, under certain additional assumptions on the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113028.png" />, showed that all such functions can be expressed in terms of hypergeometric functions and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113029.png" /> satisfies a linear second-order differential equation with rational coefficients, but did not write this equation out explicitly (see [[#References|[1]]]). The equation in question, (1), was given by E. Papperitz [[#References|[2]]]. It is also called the Riemann <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113031.png" />-equation, the Riemann equation in Papperitz's form and the Riemann equation, and its solutions are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113033.png" />-functions.
+
Riemann, under certain additional assumptions on the numbers $  \alpha , \alpha  ^  \prime  \dots \gamma  ^  \prime  $,  
 +
showed that all such functions can be expressed in terms of hypergeometric functions and that $  w( z) $
 +
satisfies a linear second-order differential equation with rational coefficients, but did not write this equation out explicitly (see [[#References|[1]]]). The equation in question, (1), was given by E. Papperitz [[#References|[2]]]. It is also called the Riemann $  P $-
 +
equation, the Riemann equation in Papperitz's form and the Riemann equation, and its solutions are called $  P $-
 +
functions.
  
 
The basic properties of the solutions of a Papperitz equation are as follows.
 
The basic properties of the solutions of a Papperitz equation are as follows.
  
1) A Papperitz equation is invariant under rational-linear transformations: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113034.png" /> maps the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113035.png" /> to points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113036.png" />, then
+
1) A Papperitz equation is invariant under rational-linear transformations: If $  z _ {1} = ( Az + b)/( Cz + D) $
 +
maps the points $  a, b, c $
 +
to points $  a _ {1} , b _ {1} , c _ {1} $,  
 +
then
 +
 
 +
$$
 +
P \left \{
 +
 
 +
\begin{array}{llll}
 +
a  & b  & c  &{}  \\
 +
\alpha  &\beta  &\gamma  & z  \\
 +
\alpha  ^  \prime  &\beta  ^  \prime  &\gamma  ^  \prime  &{}  \\
 +
\end{array}
 +
\right \}
 +
=  P \left \{
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113037.png" /></td> </tr></table>
+
\begin{array}{llll}
 +
a _ {1}  &b _ {1}  &c _ {1}  &{}  \\
 +
\alpha  &\beta  &\gamma  &z _ {1}  \\
 +
\alpha  ^  \prime  &\beta  ^  \prime  &\gamma  ^  \prime  &{}  \\
 +
\end{array}
 +
\right \} .
 +
$$
  
 
2) The transformation
 
2) The transformation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113038.png" /></td> </tr></table>
+
$$
 +
\left ( z-
 +
\frac{a}{z-
 +
b} \right )  ^ {k}
 +
\left ( z-
 +
\frac{c}{z-
 +
b }\right )  ^ {l} w  = \widetilde{w}
 +
$$
  
 
transforms a Papperitz equation into a Papperitz equation with the same singular points, but with different characteristic exponents:
 
transforms a Papperitz equation into a Papperitz equation with the same singular points, but with different characteristic exponents:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113039.png" /></td> </tr></table>
+
$$
 +
\left ( z-
 +
\frac{a}{z-
 +
b }\right )  ^ {k}
 +
\left ( z-
 +
\frac{c}{z-
 +
b} \right )  ^ {l} P \left \{
 +
 
 +
\begin{array}{llll}
 +
a  & b  & c  &{}  \\
 +
\alpha  &\beta  &\gamma  & z  \\
 +
\alpha  ^  \prime  &\beta  ^  \prime  &\gamma  ^  \prime  &{}  \\
 +
\end{array}
 +
\right \} =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113040.png" /></td> </tr></table>
+
$$
 +
= \
 +
P \left \{
 +
\begin{array}{cccl}
 +
a  & b  & c  &{}  \\
 +
\alpha + k  &\beta - k- l  &\gamma + l  & z  \\
 +
\alpha
 +
^  \prime  + k  &\beta  ^  \prime  - k- l  &\gamma  ^  \prime  + l  &{}  \\
 +
\end{array}
 +
\right \} .
 +
$$
  
 
3) The [[Hypergeometric equation|hypergeometric equation]]
 
3) The [[Hypergeometric equation|hypergeometric equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113041.png" /></td> </tr></table>
+
$$
 +
z( 1- z) w  ^ {\prime\prime} + [ C - ( A+ B+ 1) z] w  ^  \prime  - ABw  = 0
 +
$$
  
 
is a special case of a Papperitz equation and it corresponds in Riemann's notation to
 
is a special case of a Papperitz equation and it corresponds in Riemann's notation to
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113042.png" /></td> </tr></table>
+
$$
 +
P \left \{
 +
 
 +
\begin{array}{clcl}
 +
0  &\infty  & 1  &{}  \\
 +
0  & A  & 0  & z  \\
 +
1- C  & B  &C- A- B  &{}  \\
 +
\end{array}
 +
\right \} .
 +
$$
  
 
4) Each solution of a Papperitz equation can be expressed in terms of the hypergeometric function,
 
4) Each solution of a Papperitz equation can be expressed in terms of the hypergeometric function,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
w( z)  = \left ( z-  
 +
\frac{a}{z-
 +
b }\right )  ^  \alpha  \left ( z-
 +
\frac{c}{z-
 +
b }\right ) ^  \gamma  \times
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113044.png" /></td> </tr></table>
+
$$
 +
\times
 +
F \left \{ \alpha + \beta + \gamma ; \alpha + \beta
 +
^  \prime  + \gamma ; 1 + \alpha -
 +
\alpha  ^  \prime  ;
 +
\frac{( z- a)( c- b) }{( z- b)( c- a) }
 +
\right \}
 +
$$
  
under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113045.png" /> is not a negative integer. If none of the differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113048.png" /> are integers, then interchanging in (2) the positions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113050.png" /> or of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113052.png" />, four solutions of a Papperitz equation are obtained. In addition a Papperitz equation remains unchanged if the positions of the triples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113055.png" /> are rearranged; all these rearrangements provide 24 special solutions of a Papperitz equation (1), which were first obtained by E.E. Kummer [[#References|[5]]].
+
under the assumption that $  \alpha - \alpha  ^  \prime  $
 +
is not a negative integer. If none of the differences $  \alpha - \alpha  ^  \prime  $,  
 +
$  \beta - \beta  ^  \prime  $,  
 +
$  \gamma - \gamma  ^  \prime  $
 +
are integers, then interchanging in (2) the positions of $  \alpha $
 +
and $  \alpha  ^  \prime  $
 +
or of $  \gamma $
 +
and $  \gamma  ^  \prime  $,  
 +
four solutions of a Papperitz equation are obtained. In addition a Papperitz equation remains unchanged if the positions of the triples $  ( \alpha , \alpha  ^  \prime  , a) $,
 +
$  ( \beta , \beta  ^  \prime  , b) $,
 +
$  ( \gamma , \gamma  ^  \prime  , c) $
 +
are rearranged; all these rearrangements provide 24 special solutions of a Papperitz equation (1), which were first obtained by E.E. Kummer [[#References|[5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann, "Beiträge zur Theorie der durch Gauss'sche Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113056.png" /> darstellbare Functionen" , ''Gesammelte math. Werke'' , Dover, reprint (1953) pp. 67–85</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Papperitz, "Ueber verwandte <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113057.png" />-Functionen" ''Math. Ann.'' , '''25''' (1885) pp. 212–221 {{MR|1510304}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 {{MR|1424469}} {{MR|0595076}} {{MR|0178117}} {{MR|1519757}} {{ZBL|0951.30002}} {{ZBL|0108.26903}} {{ZBL|0105.26901}} {{ZBL|53.0180.04}} {{ZBL|47.0190.17}} {{ZBL|45.0433.02}} {{ZBL|33.0390.01}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) {{MR|0100119}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.E. Kummer, "Ueber die hypergeometrische Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113058.png" />" ''J. Reine Angew. Math.'' , '''15''' (1836) pp. 39–83; 127–172</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B. Riemann, "Beiträge zur Theorie der durch Gauss'sche Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113056.png" /> darstellbare Functionen" , ''Gesammelte math. Werke'' , Dover, reprint (1953) pp. 67–85</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Papperitz, "Ueber verwandte <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113057.png" />-Functionen" ''Math. Ann.'' , '''25''' (1885) pp. 212–221 {{MR|1510304}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 {{MR|1424469}} {{MR|0595076}} {{MR|0178117}} {{MR|1519757}} {{ZBL|0951.30002}} {{ZBL|0108.26903}} {{ZBL|0105.26901}} {{ZBL|53.0180.04}} {{ZBL|47.0190.17}} {{ZBL|45.0433.02}} {{ZBL|33.0390.01}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) {{MR|0100119}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.E. Kummer, "Ueber die hypergeometrische Reihe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071130/p07113058.png" />" ''J. Reine Angew. Math.'' , '''15''' (1836) pp. 39–83; 127–172</TD></TR></table>

Latest revision as of 08:38, 14 July 2022


An ordinary second-order Fuchsian linear differential equation having precisely three singular points:

$$ \tag{1 } w ^ {\prime\prime } + \left ( \frac{1 - \alpha - \alpha ^ \prime }{z- a} + \frac{1 - \beta - \beta ^ \prime }{z- b} + \frac{1- \gamma - \gamma ^ \prime }{z- c} \right ) w ^ \prime + $$

$$ + \left [ \frac{\alpha \alpha ^ \prime ( a- b)( a- c) }{z- a} + \frac{\beta \beta ^ \prime ( b- c)( b- a) }{z- b}\right . + $$

$$ + \left . \frac{\gamma \gamma ^ \prime ( c- a)( c- b) }{z- c} \right ] \frac{w}{(z- a)( z- b)( z- c)} = 0 , $$

$$ \alpha + \alpha ^ \prime + \beta + \beta ^ \prime + \gamma + \gamma ^ \prime = 1; $$

here $ a, b, c $ are pairwise distinct complex numbers, $ \alpha , \alpha ^ \prime $( $ \beta , \beta ^ \prime $ and $ \gamma , \gamma ^ \prime $) are the characteristic exponents at the singular point $ z= a $( respectively, $ z= b $ and $ z= c $). A Papperitz equation is uniquely determined by the assignment of the singular points and the characteristic exponents. In solving a Papperitz equation (1), use is made of Riemann's notation:

$$ w = P \left \{ \begin{array}{llll} a & b & c &{} \\ \alpha &\beta &\gamma & z \\ \alpha ^ \prime &\beta ^ \prime &\gamma ^ \prime &{} \\ \end{array} \right \} . $$

B. Riemann investigated [1] the problem of finding all many-valued functions $ w( z) $, analytic in the extended complex plane, which have the following properties:

a) the function $ w( z) $ has precisely three singular points $ a, b, c $;

b) any three of its branches are connected by a linear equation

$$ A _ {1} w _ {1} ( z) + A _ {2} w _ {2} ( z) + A _ {3} w _ {3} ( z) = 0 $$

with constant coefficients;

c) the function $ w( z) $ has the simplest singularities at the points $ a, b, c $; namely, in a neighbourhood of the point $ z= a $ there are two branches $ \widetilde{w} _ {1} ( z) $ and $ \widetilde{w} _ {2} ( z) $ satisfying

$$ \widetilde{w} _ {1} ( z) = \ ( z- a) ^ \alpha \phi _ {1} ( z) ,\ \ \widetilde{w} _ {2} ( z) = \ ( z- a) ^ {\alpha ^ \prime } \phi _ {2} ( z) , $$

where $ \phi _ {j} ( z) $ $ ( j = 1, 2) $ is holomorphic at $ z= a $; and analogously for $ b $ and $ c $.

Riemann, under certain additional assumptions on the numbers $ \alpha , \alpha ^ \prime \dots \gamma ^ \prime $, showed that all such functions can be expressed in terms of hypergeometric functions and that $ w( z) $ satisfies a linear second-order differential equation with rational coefficients, but did not write this equation out explicitly (see [1]). The equation in question, (1), was given by E. Papperitz [2]. It is also called the Riemann $ P $- equation, the Riemann equation in Papperitz's form and the Riemann equation, and its solutions are called $ P $- functions.

The basic properties of the solutions of a Papperitz equation are as follows.

1) A Papperitz equation is invariant under rational-linear transformations: If $ z _ {1} = ( Az + b)/( Cz + D) $ maps the points $ a, b, c $ to points $ a _ {1} , b _ {1} , c _ {1} $, then

$$ P \left \{ \begin{array}{llll} a & b & c &{} \\ \alpha &\beta &\gamma & z \\ \alpha ^ \prime &\beta ^ \prime &\gamma ^ \prime &{} \\ \end{array} \right \} = P \left \{ \begin{array}{llll} a _ {1} &b _ {1} &c _ {1} &{} \\ \alpha &\beta &\gamma &z _ {1} \\ \alpha ^ \prime &\beta ^ \prime &\gamma ^ \prime &{} \\ \end{array} \right \} . $$

2) The transformation

$$ \left ( z- \frac{a}{z- b} \right ) ^ {k} \left ( z- \frac{c}{z- b }\right ) ^ {l} w = \widetilde{w} $$

transforms a Papperitz equation into a Papperitz equation with the same singular points, but with different characteristic exponents:

$$ \left ( z- \frac{a}{z- b }\right ) ^ {k} \left ( z- \frac{c}{z- b} \right ) ^ {l} P \left \{ \begin{array}{llll} a & b & c &{} \\ \alpha &\beta &\gamma & z \\ \alpha ^ \prime &\beta ^ \prime &\gamma ^ \prime &{} \\ \end{array} \right \} = $$

$$ = \ P \left \{ \begin{array}{cccl} a & b & c &{} \\ \alpha + k &\beta - k- l &\gamma + l & z \\ \alpha ^ \prime + k &\beta ^ \prime - k- l &\gamma ^ \prime + l &{} \\ \end{array} \right \} . $$

3) The hypergeometric equation

$$ z( 1- z) w ^ {\prime\prime} + [ C - ( A+ B+ 1) z] w ^ \prime - ABw = 0 $$

is a special case of a Papperitz equation and it corresponds in Riemann's notation to

$$ P \left \{ \begin{array}{clcl} 0 &\infty & 1 &{} \\ 0 & A & 0 & z \\ 1- C & B &C- A- B &{} \\ \end{array} \right \} . $$

4) Each solution of a Papperitz equation can be expressed in terms of the hypergeometric function,

$$ \tag{2 } w( z) = \left ( z- \frac{a}{z- b }\right ) ^ \alpha \left ( z- \frac{c}{z- b }\right ) ^ \gamma \times $$

$$ \times F \left \{ \alpha + \beta + \gamma ; \alpha + \beta ^ \prime + \gamma ; 1 + \alpha - \alpha ^ \prime ; \frac{( z- a)( c- b) }{( z- b)( c- a) } \right \} $$

under the assumption that $ \alpha - \alpha ^ \prime $ is not a negative integer. If none of the differences $ \alpha - \alpha ^ \prime $, $ \beta - \beta ^ \prime $, $ \gamma - \gamma ^ \prime $ are integers, then interchanging in (2) the positions of $ \alpha $ and $ \alpha ^ \prime $ or of $ \gamma $ and $ \gamma ^ \prime $, four solutions of a Papperitz equation are obtained. In addition a Papperitz equation remains unchanged if the positions of the triples $ ( \alpha , \alpha ^ \prime , a) $, $ ( \beta , \beta ^ \prime , b) $, $ ( \gamma , \gamma ^ \prime , c) $ are rearranged; all these rearrangements provide 24 special solutions of a Papperitz equation (1), which were first obtained by E.E. Kummer [5].

References

[1] B. Riemann, "Beiträge zur Theorie der durch Gauss'sche Reihe darstellbare Functionen" , Gesammelte math. Werke , Dover, reprint (1953) pp. 67–85
[2] E. Papperitz, "Ueber verwandte -Functionen" Math. Ann. , 25 (1885) pp. 212–221 MR1510304
[3] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952) pp. Chapt. 6 MR1424469 MR0595076 MR0178117 MR1519757 Zbl 0951.30002 Zbl 0108.26903 Zbl 0105.26901 Zbl 53.0180.04 Zbl 47.0190.17 Zbl 45.0433.02 Zbl 33.0390.01
[4] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119
[5] E.E. Kummer, "Ueber die hypergeometrische Reihe " J. Reine Angew. Math. , 15 (1836) pp. 39–83; 127–172
How to Cite This Entry:
Papperitz equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Papperitz_equation&oldid=24524
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article