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)