View source for Hypergeometric equation

''Gauss equation''

An ordinary second-order linear differential 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/h/h048/h048440/h0484401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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or, in self-adjoint form,

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The variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h0484404.png" /> and the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h0484405.png" /> assume, in the general case, complex values. After substituting

<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/h/h048/h048440/h0484406.png" /></td> </tr></table>

the reduced form of equation (1) is obtained:

<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/h/h048/h048440/h0484407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h0484408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h0484409.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844010.png" />.

Equation (1) was studied in detail by C.F. Gauss

in connection with his theory of the [[Hypergeometric series|hypergeometric series]], but had been considered (together with its solution) by L. Euler at an even earlier date.

Solutions of equation (1) are expressed by way of the [[Hypergeometric function|hypergeometric function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844012.png" /> is not an integer, the general solution of (1) may be written as

<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/h/h048/h048440/h04844013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

<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/h/h048/h048440/h04844014.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844016.png" /> are arbitrary constants. The representation (3) is valid in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844017.png" />-plane with slits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844019.png" />. In particular, in the real case (3) yields the general solution of (1) in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844020.png" />. For integral values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844021.png" /> the general solution is more complicated (the individual terms may contain logarithms).

Functions other than those shown in (3) may also be selected as a [[Fundamental system of solutions|fundamental system of solutions]] of equation (1). For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844022.png" /> is not an integer, then

<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/h/h048/h048440/h04844023.png" /></td> </tr></table>

<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/h/h048/h048440/h04844024.png" /></td> </tr></table>

is the general solution of (1) in the complex plane with slit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844025.png" /> [[#References|[2]]], [[#References|[3]]].

The Gauss equations include, as particular cases, a number of differential equations encountered in applications; many ordinary linear second-order differential equations are reduced to (1) by transforming the unknown function and the independent variable [[#References|[4]]]. The [[Confluent hypergeometric equation|confluent hypergeometric equation]], which is close to equation (1), is particularly important. The ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844026.png" /> of two linearly independent solutions of equation (2) satisfies the [[Schwarz equation|Schwarz equation]], which is closely connected with the problem of conformal mapping a semi-plane onto a triangle bounded by three peripheral arcs. The study of the inverse function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844027.png" /> leads to the concept of an [[Automorphic function|automorphic function]] [[#References|[5]]].

There exist linear equations of higher orders whose solutions display properties similar to those of hypergeometric functions: The solution of the following equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844028.png" />,

<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/h/h048/h048440/h04844029.png" /></td> </tr></table>

is the generalized hypergeometric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844030.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844031.png" /> parameters. In particular, the generalized hypergeometric equation of the third order, the solution of which is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844032.png" />, may be represented as

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====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  C.F. Gauss,   "Disquisitiones generales circa seriem infinitam <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844036.png" />"  ''Comm. Soc. Regia Sci. Göttingen Rec.'' , '''2'''  (1812)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  C.F. Gauss,   "Disquisitiones generales circa seriem infinitam <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844037.png" />" , ''Werke'' , '''3''' , K. Gesellschaft Wissenschaft. Göttingen  (1876)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Krazer,   W. Franz,   "Transzendente Funktionen" , Akademie Verlag  (1960)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1–3''' , McGraw-Hill  (1953–1955)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1971)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.V. Golubev,   "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR></table>



====Comments====
The hypergeometric equation is a differential equation with three regular singular points (cf. [[Regular singular point|Regular singular point]]) at 0, 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844038.png" /> such that both at 0 and 1 one of the exponents equals 0. So it is a special case of the [[Riemann differential equation|Riemann differential equation]]. The hypergeometric equation has been generalized to a system of partial differential equations with regular singularities such that the Appell or Lauricella hypergeometric function in several variables is a solution, cf. [[#References|[a1]]]. In [[#References|[a2]]], [[#References|[a3]]] a study is made of a second-order partial differential equation associated with a root system, which generalizes the case of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048440/h04844039.png" /> for the ordinary hypergeometric equation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Appell,   M.J. Kampé de Fériet,   "Fonctions hypergéométriques et hypersphériques: Polynômes d'Hermite" , Gauthier-Villars  (1926)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.J. Heckman,   E.M. Opdam,   "Root systems and hypergeometric functions I"  ''Compositio Math.'' , '''64'''  (1987)  pp. 329–352</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.J. Heckman,   "Root systems and hypergeometric functions II"  ''Compositio Math.'' , '''64'''  (1987)  pp. 353–373</TD></TR></table>