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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 ∈ C {\displaystyle a_{1},\dots ,a_{r}\in \mathbb {C} } be the r {\displaystyle r} regular singularities in the finite part of the complex plane of the linear differential equation L f := d n f d z n + q 1 d n − 1 f d z n − 1 + ⋯ + q n − 1 d f d z + q n f {\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 q i {\displaystyle q_{i}} . For linear differential equations the singularities are exactly the singular points of the coefficients. L f = 0 {\displaystyle Lf=0} is a Fuchsian equation if and only if the coefficients are rational functions of the form

q i ( z ) = Q i ( z ) ψ i {\displaystyle q_{i}(z)={\frac {Q_{i}(z)}{\psi ^{i}}}}

with the polynomial ψ := ∏ j = 0 r ( z − a j ) ∈ C [ z ] {\textstyle \psi :=\prod _{j=0}^{r}(z-a_{j})\in \mathbb {C} [z]} and certain polynomials Q i ∈ C [ z ] {\displaystyle Q_{i}\in \mathbb {C} [z]} for i ∈ { 1 , … , n } {\displaystyle i\in \{1,\dots ,n\}} , such that deg ⁡ ( Q i ) ≤ i ( r − 1 ) {\displaystyle \deg(Q_{i})\leq i(r-1)} .² This means the coefficient q i {\displaystyle q_{i}} has poles of order at most i {\displaystyle i} , for i ∈ { 1 , … , n } {\displaystyle i\in \{1,\dots ,n\}} .

Fuchs relation

Let L f = 0 {\displaystyle Lf=0} be a Fuchsian equation of order n {\displaystyle n} with the singularities a 1 , … , a r ∈ C {\displaystyle a_{1},\dots ,a_{r}\in \mathbb {C} } and the point at infinity. Let α i 1 , … , α i n ∈ C {\displaystyle \alpha _{i1},\dots ,\alpha _{in}\in \mathbb {C} } be the roots of the indicial polynomial relative to a i {\displaystyle a_{i}} , for i ∈ { 1 , … , r } {\displaystyle i\in \{1,\dots ,r\}} . Let β 1 , … , β n ∈ C {\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 L f {\displaystyle Lf} transformed by z = x − 1 {\displaystyle z=x^{-1}} at x = 0 {\displaystyle x=0} . Then the so called Fuchs relation holds:

∑ i = 1 r ∑ k = 1 n α i k + ∑ k = 1 n β k = n ( n − 1 ) ( r − 1 ) 2 {\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 P ξ {\displaystyle P_{\xi }} denote the indicial polynomial relative to ξ ∈ C ∪ { ∞ } {\displaystyle \xi \in \mathbb {C} \cup \{\infty \}} of the Fuchsian equation L f = 0 {\displaystyle Lf=0} . Define defect : C ∪ { ∞ } → C {\displaystyle \operatorname {defect} :\mathbb {C} \cup \{\infty \}\to \mathbb {C} } as

defect ⁡ ( ξ ) := { Tr ⁡ ( P ξ ) − n ( n − 1 ) 2 , for  ξ ∈ C Tr ⁡ ( P ξ ) + n ( n − 1 ) 2 , for  ξ = ∞ {\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 Tr ⁡ ( P ) := ∑ { z ∈ C : P ( z ) = 0 } z {\textstyle \operatorname {Tr} (P):=\sum _{\{z\in \mathbb {C} :P(z)=0\}}z} gives the trace of a polynomial P {\displaystyle P} , i. e., Tr {\displaystyle \operatorname {Tr} } denotes the sum of a polynomial's roots counted with multiplicity.

This means that defect ⁡ ( ξ ) = 0 {\displaystyle \operatorname {defect} (\xi )=0} for any ordinary point ξ {\displaystyle \xi } , due to the fact that the indicial polynomial relative to any ordinary point is P ξ ( α ) = α ( α − 1 ) ⋯ ( α − n + 1 ) {\displaystyle P_{\xi }(\alpha )=\alpha (\alpha -1)\cdots (\alpha -n+1)} . The transformation z = x − 1 {\displaystyle z=x^{-1}} , that is used to obtain the indicial equation relative to ∞ {\displaystyle \infty } , motivates the changed sign in the definition of defect {\displaystyle \operatorname {defect} } for ξ = ∞ {\displaystyle \xi =\infty } . The rewritten Fuchs relation is:

∑ ξ ∈ C ∪ { ∞ } defect ⁡ ( ξ ) = 0. {\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.

References

1. Ince, Edward Lindsay (1956). Ordinary Differential Equations. New York, USA: Dover Publications. p. 370. ISBN 9780486158211. {{cite book}}: ISBN / Date incompatibility (help) 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. {{cite book}}: ISBN / Date incompatibility (help) 9780486158211 ↩

4. Landl, Elisabeth (2018). The Fuchs Relation (Bachelor Thesis). Linz, Austria. chapter 3. ↩