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

with meromorphic functions $$q_i$$. For linear differential equations the singularities are exactly the singular points of the coefficients. $$Lf=0$$ is a Fuchsian equation if and only if the coefficients are rational functions of the form


 * $$q_i(z) = \frac{Q_i(z)}{\psi^i}$$

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

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


 * $$\sum_{i=1}^r \sum_{k=1}^n \alpha_{ik} + \sum_{k=1}^n \beta_{k} = \frac{n(n-1)(r-1)}{2}$$.

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


 * $$\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}$$

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

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


 * $$\sum_{\xi\in\mathbb{C}\cup\{\infty\}} \operatorname{defect}(\xi) = 0.$$