Rank of an ordinary linear differential equation

in the complex domain,

$$\tag{1 } \sum_{j=0} ^ { n } P _ {j} ( z) w ^ {( n - j ) } = 0 ,\ \ P _ {0} ( z) = 1$$

The number $r = k + 1$, where

$$k = \max _ {1 \leq j \leq n } \frac{n _ {j} }{j} ,$$

if the coefficients in equation

are series which are convergent for large $| z |$:

$$P _ {j} ( z) = \ \sum _ { m = - \infty } ^ { {n _ j } } p _ {jm} z ^ {m} ,\ \ j = 1 \dots n .$$

The concept of rank is used only when $z = \infty$ is a singular point of the differential equation . The rank of the differential equation is also called the rank of the singular point $z = \infty$. If this point is a regular singular point, then $r = 0$; if it is an irregular singular point, then $r > 0$. The number $k$ is called the subrank. The rank of the equation is an integer or a rational number. If the subrank is rational with denominator $q \geq 2$, then the subrank of the equation obtained from

by the change of variable $z = \zeta ^ {q}$ is an integer. The rank of the equation is invariant with respect to a change of variable of the form $z = \zeta \phi ( \zeta )$, with $\phi$ a holomorphic function at the point $\zeta = \infty$ that is non-zero at this point.

The concept of the rank of an equation is used in investigating the structure of the solutions to equation

with a singular point at infinity. Let $Q ( z)$ be a polynomial of degree $p$, let

$$\Psi ( \zeta ) = \sum_{m=0} ^ \infty \psi _ {m} \zeta ^ {-m}$$

be a formal series, and let $s \geq 1$ be an integer. The series

$$\tag{2 } w = e ^ {Q ( z ^ {1/s} ) } z ^ \rho \Psi ( z ^ {1/s} )$$

is a normal (subnormal, respectively) series of order $p / s$ if $s = 1$( $s \geq 2$, respectively). A solution to equation

which is represented by a normal (subnormal) series, convergent in a neighbourhood of $z = \infty$, is called a normal (subnormal) solution of the same order (see [2], [3]).

The order of a normal (subnormal) solution does not exceed the rank of the equation; this is true also for formal solutions of the form (2). If the rank $r$ of equation

is an integer, then

has at least one formal solution of the form (2) of order $r$. The substitution $w ( z) = e ^ {Q ( z) } u ( z)$ does not alter the rank of the equation. If the subrank is $k = p / q$, where $p , q$ are mutually prime integers and $q \geq 2$, then the equation has no less than $q$ formal solutions of the form (2) of order $r$.

A Hamburger equation is an equation

with rational coefficients which has exactly two singular points: a regular one $z = 0$ and an irregular one $z = \infty$. For a Hamburger equation one can obtain sufficient conditions for it to have normal solutions; when $n = 2$ there are necessary and sufficient conditions for the existence of normal and subnormal solutions (see [2]).

One also introduces the concept of rank in the case when equation

has a finite singular point (see [2], [3]).

In the case of a linear system of $n$ ordinary differential equations in the complex domain,

$$\tag{3 } w ^ \prime = z ^ {r} A ( z) w ,$$

where $r \geq - 1$ is an integer and the matrix-function $A ( z)$ is holomorphic at $z = \infty$ and $A ( \infty ) \neq 0$, the number $r + 1$ is called the rank of the system (3) or the rank of the singular point $z = \infty$, the number $r$ is its subrank (see [4]–[6]). If $r = - 1$, then the point $z = \infty$ is a regular singular point; in contrast to a scalar equation , the point $z = \infty$ can be a regular singular point if $r > - 1$( see [4]).

References

[1]	H. Poincaré, "Sur les intégrales irregulières des équations linéaires" Acta Math. , 8 (1866) pp. 295–344
[2]	E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
[3]	K.Ya. Latysheva, N.I. Tereshchenko, G.S. Orel, "Normally regular solutions and their applications" , Kiev (1974) (In Russian)
[4]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[5]	E. Kamke, "Handbuch der gewöhnliche Differentialgleichungen" , Chelsea, reprint (1947)
[6]	W. Wasov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)

Comments

For the word rank sometimes the term grade is used. The following can be proved, see [a1]. For every $r$ a solution $w( z)$ exists for which

$$\lim\limits _ {| z | \rightarrow \infty } | z | ^ {-} r \mathop{\rm log} | w( z) | > 0 ,$$

except for a finite number of directions.

References

[a1]	E. Hille, "Lectures on ordinary differential equations" , Addison-Wesley (1969)