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 $a_1, \dots, a_r \in \mathbb$ be the $r$ regular singularities in the finite part of the complex plane of the linear differential equation$Lf := \frac + q_1\frac + \cdots + q_\frac + q_nf$

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

: $q_i(z) = \frac$

with the polynomial \psi := \prod_^r (z-a_j) \in\mathbb /math> and certain polynomials Q_i \in \mathbb /math> for $i\in \$, 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 \$.

Fuchs relation

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

: $\sum_^r \sum_^n \alpha_ + \sum_^n \beta_ = \frac$.

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

: $\operatorname(\xi):= \begin \operatorname(P_\xi) - \frac\text\xi\in\mathbb\\ \operatorname(P_\xi) + \frac\text\xi=\infty \end$

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

This means that $\operatorname(\xi)=0$ 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 $z=x^$, that is used to obtain the indicial equation relative to $\infty$, motivates the changed sign in the definition of $\operatorname$ for $\xi=\infty$. The rewritten Fuchs relation is:

: $\sum_ \operatorname(\xi) = 0.$Landl, Elisabeth (2018). The Fuchs Relation (Bachelor Thesis). Linz, Austria. chapter 3.

References

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* {{Cite book, title=Handbuch der Theorie der linearen Differentialgleichungen (2. Band, 1. Teil), last=Schlesinger, first=Ludwig, publisher=B. G.Teubner, year=1897, isbn=, location=Leipzig, Germany, pages=241 ff

Complex analysis
Differential equations