# Hypergeometric equation

Gauss equation

An ordinary second-order linear differential equation

$$\tag{1 } z ( z - 1) w ^ {\prime\prime} + [( \alpha + \beta + 1) z - \gamma ] w ^ \prime + \alpha \beta w = 0,$$

$$\alpha , \beta , \gamma = \textrm{ const },$$

$$[ z ^ \gamma ( z - 1) ^ {\alpha + \beta + 1 - \gamma } w ^ \prime ] ^ \prime + \alpha \beta z ^ {\gamma - 1 } ( z - 1) ^ {\alpha + \beta - \gamma } w = 0.$$

The variables $z, w$ and the parameters $\alpha , \beta , \gamma$ assume, in the general case, complex values. After substituting

$$w = z ^ {- \gamma / 2 } ( z - 1) ^ {( \gamma - \alpha - \beta - 1)/2 } u$$

the reduced form of equation (1) is obtained:

$$\tag{2 } u ^ {\prime\prime} + \left [ \frac{1 - \lambda ^ {2} }{4z ^ {2} } + \frac{1 - \gamma ^ {2} }{4 ( z - 1) ^ {2} } - \frac{1 - \lambda ^ {2} + \mu ^ {2} - \gamma ^ {2} }{4z ( z - 1) } \right ] u = 0,$$

where $\lambda = 1 - \gamma$, $\mu = \alpha - \beta$, and $\nu = \gamma - \alpha - \beta$.

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

in connection with his theory of the 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 $F( \alpha , \beta ; \gamma ; z )$. If $\gamma$ is not an integer, the general solution of (1) may be written as

$$\tag{3 } w = C _ {1} F ( \alpha , \beta ; \gamma ; z) +$$

$$+ C _ {2} z ^ {1 - \gamma } F ( \alpha - \gamma + 1, \beta - \gamma + 1; 2 - \gamma ; z),$$

where $C _ {1}$ and $C _ {2}$ are arbitrary constants. The representation (3) is valid in the complex $z$- plane with slits $( - \infty , 0)$ and $( 1, \infty )$. In particular, in the real case (3) yields the general solution of (1) in the interval $0 < z < 1$. For integral values of $\gamma$ 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 of equation (1). For instance, if $\alpha - \beta$ is not an integer, then

$$w = C _ {1} (- z) ^ {- \alpha } F ( \alpha , \alpha - \gamma + 1; \ \alpha - \beta + 1; z ^ {- 1 } ) +$$

$$+ C _ {2} (- z) ^ {- \beta } F ( \beta - \gamma + 1, \beta ; \beta - \alpha + 1; z ^ {- 1 } )$$

is the general solution of (1) in the complex plane with slit $( 0, \infty )$, .

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 . The confluent hypergeometric equation, which is close to equation (1), is particularly important. The ratio $s( z)$ of two linearly independent solutions of equation (2) satisfies the 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 $z( s)$ leads to the concept of an automorphic function .

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 $q+ 1$,

$$\left [ z { \frac{d}{dz} } \prod _ {j = 1 } ^ { q } \left ( z { \frac{d}{dz} } + \gamma _ {j} - 1 \right ) - z \prod _ {i = 1 } ^ { p } \left ( z { \frac{d}{dz} } + \alpha _ {i} \right ) \right ] w = 0,$$

is the generalized hypergeometric function ${} _ {p} F _ {q} ( \alpha _ {i} ; \gamma _ {j} ; z)$ with $p+ q$ parameters. In particular, the generalized hypergeometric equation of the third order, the solution of which is ${} _ {3} F _ {2} ( \alpha _ {1} , \alpha _ {2} , \alpha _ {3} ; \gamma _ {1} , \gamma _ {2} ; z )$, may be represented as

$$z ^ {2} ( 1 - z) w ^ {\prime\prime\prime} + [ 1 + \gamma _ {1} + \gamma _ {2} - ( 3 + \alpha _ {1} + \alpha _ {2} + \alpha _ {3} ) z] zw ^ {\prime\prime} +$$

$$+ [ \gamma _ {1} \gamma _ {2} - ( 1 + \alpha _ {1} + \alpha _ {2} + \alpha _ {3} + \alpha _ {1} \alpha _ {2} + \alpha _ {2} \alpha _ {3} + \alpha _ {3} \alpha _ {1} ) z] w ^ \prime +$$

$$- \alpha _ {1} \alpha _ {2} \alpha _ {3} w = 0.$$

How to Cite This Entry:
Hypergeometric equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypergeometric_equation&oldid=47297
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article