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

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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

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Let   be the   regular singularities in the finite part of the complex plane of the linear differential equation 

with meromorphic functions  . For linear differential equations the singularities are exactly the singular points of the coefficients.   is a Fuchsian equation if and only if the coefficients are rational functions of the form

 

with the polynomial   and certain polynomials   for  , such that  .[2] This means the coefficient   has poles of order at most  , for  .

Fuchs relation

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Let   be a Fuchsian equation of order   with the singularities   and the point at infinity. Let   be the roots of the indicial polynomial relative to  , for  . Let   be the roots of the indicial polynomial relative to  , which is given by the indicial polynomial of   transformed by   at  . Then the so called Fuchs relation holds:

 .[3]

The Fuchs relation can be rewritten as infinite sum. Let   denote the indicial polynomial relative to   of the Fuchsian equation  . Define   as

 

where   gives the trace of a polynomial  , i. e.,   denotes the sum of a polynomial's roots counted with multiplicity.

This means that   for any ordinary point  , due to the fact that the indicial polynomial relative to any ordinary point is  . The transformation  , that is used to obtain the indicial equation relative to  , motivates the changed sign in the definition of   for  . The rewritten Fuchs relation is:

 [4]

References

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