Fuchs relation

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (May 2025) (Learn how and when to remove this message) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (May 2025) (Learn how and when to remove this message) (Learn how and when to remove this message)

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.

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.

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]

• Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. ISBN 9780486158211. {{cite book}}: ISBN / Date incompatibility (help)
• Tenenbaum, Morris; Pollard, Harry (1963). Ordinary Differential Equations. New York, USA: Dover Publications. pp. Lecture 40. ISBN 9780486649405. {{cite book}}: ISBN / Date incompatibility (help)
• 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.