Fuchs relation

In mathematics, the Fuchs relation is a relation between the starting exponents of formal series solutions of certain linear differential equations, so called Fuchsian equations. It is named after Lazarus Immanuel Fuchs.

Definition Fuchsian equation

A linear differential equation in which every singular point, including the point at infinity, is a regular singularity is called Fuchsian equation or equation of Fuchsian type. For Fuchsian equations a formal fundamental system exists at any point, due to the Fuchsian theory.

Coefficients of a Fuchsian equation

Let [math]\displaystyle{ a_1, \dots, a_r \in \mathbb{C} }[/math] be the [math]\displaystyle{ r }[/math] regular singularities in the finite part of the complex plane of the linear differential equation[math]\displaystyle{ Lf := \frac{d^nf}{dz^n} + q_1\frac{d^{n-1}f}{dz^{n-1}} + \cdots + q_{n-1}\frac{df}{dz} + q_nf }[/math]

with meromorphic functions [math]\displaystyle{ q_i }[/math]. For linear differential equations the singularities are exactly the singular points of the coefficients. [math]\displaystyle{ Lf=0 }[/math] is a Fuchsian equation if and only if the coefficients are rational functions of the form

[math]\displaystyle{ q_i(z) = \frac{Q_i(z)}{\psi^i} }[/math]

with the polynomial [math]\displaystyle{ \psi := \prod_{j=0}^r (z-a_j) \in\mathbb{C}[z] }[/math] and certain polynomials [math]\displaystyle{ Q_i \in \mathbb{C}[z] }[/math] for [math]\displaystyle{ i\in \{1,\dots,n\} }[/math], such that [math]\displaystyle{ \deg(Q_i) \leq i(r-1) }[/math]. This means the coefficient [math]\displaystyle{ q_i }[/math] has poles of order at most [math]\displaystyle{ i }[/math], for [math]\displaystyle{ i\in \{1,\dots,n\} }[/math].

Fuchs relation

Let [math]\displaystyle{ Lf=0 }[/math] be a Fuchsian equation of order [math]\displaystyle{ n }[/math] with the singularities [math]\displaystyle{ a_1, \dots, a_r\in\mathbb{C} }[/math] and the point at infinity. Let [math]\displaystyle{ \alpha_{i1},\dots,\alpha_{in}\in\mathbb{C} }[/math] be the roots of the indicial polynomial relative to [math]\displaystyle{ a_i }[/math], for [math]\displaystyle{ i\in\{1,\dots,r\} }[/math]. Let [math]\displaystyle{ \beta_1,\dots,\beta_n\in\mathbb{C} }[/math] be the roots of the indicial polynomial relative to [math]\displaystyle{ \infty }[/math], which is given by the indicial polynomial of [math]\displaystyle{ Lf }[/math] transformed by [math]\displaystyle{ z=x^{-1} }[/math] at [math]\displaystyle{ x=0 }[/math]. Then the so called Fuchs relation holds:

[math]\displaystyle{ \sum_{i=1}^r \sum_{k=1}^n \alpha_{ik} + \sum_{k=1}^n \beta_{k} = \frac{n(n-1)(r-1)}{2} }[/math].

The Fuchs relation can be rewritten as infinite sum. Let [math]\displaystyle{ P_{\xi} }[/math] denote the indicial polynomial relative to [math]\displaystyle{ \xi\in\mathbb{C}\cup\{\infty\} }[/math] of the Fuchsian equation [math]\displaystyle{ Lf=0 }[/math]. Define [math]\displaystyle{ \operatorname{defect}: \mathbb{C}\cup\{\infty\}\to\mathbb{C} }[/math] as

[math]\displaystyle{ \operatorname{defect}(\xi):= \begin{cases} \operatorname{Tr}(P_\xi) - \frac{n(n-1)}{2}\text{, for }\xi\in\mathbb{C}\\ \operatorname{Tr}(P_\xi) + \frac{n(n-1)}{2}\text{, for }\xi=\infty \end{cases} }[/math]

where [math]\displaystyle{ \operatorname{Tr}(P):=\sum_{\{z\in\mathbb{C}: P(z)=0\}} z }[/math] gives the trace of a polynomial [math]\displaystyle{ P }[/math], i. e., [math]\displaystyle{ \operatorname{Tr} }[/math] denotes the sum of a polynomial's roots counted with multiplicity.

This means that [math]\displaystyle{ \operatorname{defect}(\xi)=0 }[/math] for any ordinary point [math]\displaystyle{ \xi }[/math], due to the fact that the indicial polynomial relative to any ordinary point is [math]\displaystyle{ P_\xi(\alpha)= \alpha(\alpha-1)\cdots(\alpha-n+1) }[/math]. The transformation [math]\displaystyle{ z=x^{-1} }[/math], that is used to obtain the indicial equation relative to [math]\displaystyle{ \infty }[/math], motivates the changed sign in the definition of [math]\displaystyle{ \operatorname{defect} }[/math] for [math]\displaystyle{ \xi=\infty }[/math]. The rewritten Fuchs relation is:

[math]\displaystyle{ \sum_{\xi\in\mathbb{C}\cup\{\infty\}} \operatorname{defect}(\xi) = 0. }[/math]

References

• Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. ISBN 9780486158211. 
• Tenenbaum, Morris; Pollard, Harry (1963). Ordinary Differential Equations. New York, USA: Dover Publications. pp. Lecture 40. ISBN 9780486649405. https://archive.org/details/ordinarydifferen00tene_0. 
• Horn, Jakob (1905). Gewöhnliche Differentialgleichungen beliebiger Ordnung. Leipzig, Germany: G. J. Göschensche Verlagshandlung. 
• Schlesinger, Ludwig (1897). Handbuch der Theorie der linearen Differentialgleichungen (2. Band, 1. Teil). Leipzig, Germany: B. G.Teubner. pp. 241 ff.