Fuchs relation explained

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

Coefficients of a Fuchsian equation

Let

a _{1,} ... , a _{r} \in C be the r regular singularities in the finite part of the complex plane of the linear differential equationLf : = (d ^{nf})/(dz ^{n}) + q _{1 (d ^{n-1} f)/(dz ^{n-1})} + … + q _{n-1} (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 formq _{i(z)} = (Q _{i(z)})/(\psi ^{i})

with the polynomial $\backslash psi\; :=\; \backslash prod\_^r\; (z-a\_j)\; \backslash in\backslash mathbb[z]$ and certain polynomials

Q _{i} \in C [z] for i\in \{ 1 , ... ,n\} , such that \deg(Q _{i)} \leq i(r -1 ) .^[2] This means the coefficient q _{i} has poles of order at most i , for i\in \{ 1 , ... ,n\} .

Fuchs relation

Let

Lf =0 be a Fuchsian equation of order n with the singularities a _{1,} ... , a _{r\inC} and the point at infinity. Let \alpha _{i1} , ... ,\alpha _{in} \in C be the roots of the indicial polynomial relative to a _{i} , for i\in\{ 1 , ... ,r\} . Let \beta _{1,...,\beta} _{n\inC} be the roots of the indicial polynomial relative to inf ty , 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 ^{r} _{i =1} \sum ^{n} _{k =1} \alpha _{ik} + \sum ^{n} _{k =1} \beta _{k} = (n(n -1 )(r -1 ))/(2) .^[3]

The Fuchs relation can be rewritten as infinite sum. Let

P _{\xi} denote the indicial polynomial relative to \xi\in C \cup\{ inf ty\} of the Fuchsian equation Lf =0 . Define \operatorname{defect}: C \cup\{ inf ty\}\to C as\operatorname{defect}(\xi): = \begin{cases} \operatorname{Tr}(P _{\xi)} - (n(n -1 ))/(2) ,for \xi\in C \\ \operatorname{Tr}(P _{\xi)} + (n(n -1 ))/(2) ,for \xi =inf ty \end{cases}

where $\backslash operatorname(P):=\backslash sum\_\; z$ 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 ) … (\alpha - n +1 ) . The transformation z = x ^{-1} , that is used to obtain the indicial equation relative to inf ty , motivates the changed sign in the definition of \operatorname{defect} for \xi =inf ty . The rewritten Fuchs relation is:\sum _{\xi\inC\cup\{infty\} } \operatorname(\xi) = 0.^[4]

References

• Book: Ince, Edward Lindsay. Ordinary Differential Equations. Dover Publications. 1956. 9780486158211. New York, USA.
• Book: Ordinary Differential Equations. Tenenbaum. Morris. Pollard. Harry. Dover Publications. 1963. 9780486649405. New York, USA. Lecture 40. registration.
• Book: Horn, Jakob. Gewöhnliche Differentialgleichungen beliebiger Ordnung. G. J. Göschensche Verlagshandlung. 1905. Leipzig, Germany.
• Book: Schlesinger, Ludwig. Handbuch der Theorie der linearen Differentialgleichungen (2. Band, 1. Teil). B. G.Teubner. 1897. Leipzig, Germany. 241 ff.

1. Book: Ince, Edward Lindsay. Ordinary Differential Equations. Dover Publications. 1956. 9780486158211. New York, USA. 370.
2. Book: Horn, Jakob. Gewöhnliche Differentialgleichungen beliebiger Ordnung. G. J. Göschensche Verlagshandlung. 1905. Leipzig, Germany. 169.
3. Book: Ince, Edward Lindsay. Ordinary Differential Equations. Dover Publications. 1956. 9780486158211. New York, USA. 371.
4. Landl, Elisabeth (2018). The Fuchs Relation (Bachelor Thesis). Linz, Austria. chapter 3.