Fuchs relation

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.^[1] For Fuchsian equations a formal fundamental system exists at any point, due to the Fuchsian theory.

Coefficients of a Fuchsian equation

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

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

${\displaystyle q_{i}(z)={\frac {Q_{i}(z)}{\psi ^{i}}}}$

with the polynomial ${\textstyle \psi$ :=\prod _{j=0}^{r}(z-a_{j})\in \mathbb {C} [z]} and certain polynomials ${\displaystyle Q_{i}\in \mathbb {C} [z]}$ for ${\displaystyle i\in \{1,\dots ,n\}}$, such that ${\displaystyle \deg(Q_{i})\leq i(r-1)}$.^[2] This means the coefficient ${\displaystyle q_{i}}$ has poles of order at most ${\displaystyle i}$, for ${\displaystyle i\in \{1,\dots ,n\}}$.

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

${\displaystyle \sum _{i=1}^{r}\sum _{k=1}^{n}\alpha _{ik}+\sum _{k=1}^{n}\beta _{k}={\frac {n(n-1)(r-1)}{2}}}$.^[3]

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

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

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

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

${\displaystyle \sum _{\xi \in \mathbb {C} \cup \{\infty \}}\operatorname {defect} (\xi )=0.}$^[4]