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$Lf:={\frac {d^{n}f}{dz^{n}}}+q_{1}{\frac {d^{n-1}f}{dz^{n-1}}}+\cdots +q_{n-1}{\frac {df}{dz}}+q_{n}f$

with meromorphic functions $q_{i}$. 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 ${\textstyle \psi :=\prod _{j=0}^{r}(z-a_{j})\in \mathbb {C} [z]}$ 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 ${\textstyle \operatorname {Tr} (P):=\sum _{\{z\in \mathbb {C} :P(z)=0\}}z}$ 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:

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

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.
• 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.