Fuchs relation

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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 a1,,ar be the r regular singularities in the finite part of the complex plane of the linear differential equationLf:=dnfdzn+q1dn1fdzn1++qn1dfdz+qnf

with meromorphic functions qi. 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

qi(z)=Qi(z)ψi

with the polynomial ψ:=j=0r(zaj)[z] and certain polynomials Qi[z] for i{1,,n}, such that deg(Qi)i(r1).[2] This means the coefficient qi has poles of order at most i, for i{1,,n}.

Fuchs relation

Let Lf=0 be a Fuchsian equation of order n with the singularities a1,,ar and the point at infinity. Let αi1,,αin be the roots of the indicial polynomial relative to ai, for i{1,,r}. Let β1,,βn be the roots of the indicial polynomial relative to , which is given by the indicial polynomial of Lf transformed by z=x1 at x=0. Then the so called Fuchs relation holds:

i=1rk=1nαik+k=1nβk=n(n1)(r1)2.[3]

The Fuchs relation can be rewritten as infinite sum. Let Pξ denote the indicial polynomial relative to ξ{} of the Fuchsian equation Lf=0. Define defect:{} as

defect(ξ):={Tr(Pξ)n(n1)2, for ξTr(Pξ)+n(n1)2, for ξ=

where Tr(P):={z:P(z)=0}z gives the trace of a polynomial P, i. e., Tr denotes the sum of a polynomial's roots counted with multiplicity.

This means that defect(ξ)=0 for any ordinary point ξ, due to the fact that the indicial polynomial relative to any ordinary point is Pξ(α)=α(α1)(αn+1). The transformation z=x1, that is used to obtain the indicial equation relative to , motivates the changed sign in the definition of defect for ξ=. The rewritten Fuchs relation is:

ξ{}defect(ξ)=0.[4]

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. https://archive.org/details/ordinarydifferen00tene_0. 
  • 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. 
  1. Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. pp. 370. ISBN 9780486158211. 
  2. Horn, Jakob (1905). Gewöhnliche Differentialgleichungen beliebiger Ordnung. Leipzig, Germany: G. J. Göschensche Verlagshandlung. pp. 169. 
  3. Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. pp. 371. ISBN 9780486158211. 
  4. Landl, Elisabeth (2018). The Fuchs Relation (Bachelor Thesis). Linz, Austria. chapter 3.