Fuchsian theory

The generalized series at $\backslash xi\backslash in\backslash mathbb\{C\}$ is defined by

: $(z-\backslash xi)^\backslash alpha\backslash sum\_\{k=0\}^\backslash infty\; c\_k(z-\backslash xi)^k,\; \backslash text\{\; with\; \}\; \backslash alpha,c\_k\; \backslash in\; \backslash mathbb\{C\}\; \backslash text\{\; and\; \}\; c\_0\backslash neq0,$

which is known as Frobenius series, due to the connection with the Frobenius series method. Frobenius series solutions are formal solutions of differential equations. The formal derivative of $z^\backslash alpha$, with $\backslash alpha\backslash in\backslash mathbb\{C\}$, is defined such that $(z^\backslash alpha)\text{'}=\backslash alpha\; z^\{\backslash alpha-1\}$. Let $f$ denote a Frobenius series relative to $\backslash xi$, then

: $\{d^nf\; \backslash over\; d\; z^n\}\; =\; (z-\backslash xi)^\{\backslash alpha-n\}\backslash sum\_\{k=0\}^\backslash infty\; (\backslash alpha+k)^\{\backslash underline\{n\}\}\; c\_k(z-\backslash xi)^k,$

where $\backslash alpha^\{\backslash underline\{n\}\}:=\backslash prod\_\{i=0\}^\{n-1\}(\backslash alpha-i)\; =\; \backslash alpha(\backslash alpha-1)\backslash cdots(\backslash alpha-n+1)$ denotes the falling factorial notation.

Let $f:=(z-\backslash xi)^\{\backslash alpha\}\backslash sum\_\{k=0\}^\{\backslash infty\}c\_k(z-\backslash xi)^k$ be a Frobenius series relative to $\backslash xi\; \backslash in\; \backslash mathbb\{C\}$. Let $Lf=f^\{(n)\}\; +\; q\_1f^\{(n-1)\}\; +\; \backslash cdots\; +\; q\_nf$

be a linear differential operator of order $n$ with one valued coefficient functions $q\_1,\; \backslash dots,\; q\_n$. Let all coefficients $q\_1,\backslash dots,q\_n$ be expandable as Laurent series with finite principle part at $\backslash xi$. Then there exists a smallest $N\backslash in\backslash mathbb\{N\}$ such that $(z-\backslash xi)^Nq\_i$ is a power series for all $i\backslash in\backslash \{1,\backslash dots,\; n\backslash \}$. Hence, $Lf$ is a Frobenius series of the form $Lf=(z-\backslash xi)^\{\backslash alpha-n-N\}\backslash psi(z)$, with a certain power series $\backslash psi(z)$ in $(z-\backslash xi)$. The indicial polynomial is defined by $P\_\{\backslash xi\}:=\backslash psi(0)$ which is a polynomial in $\backslash alpha$, i.e., $P\_\{\backslash xi\}$ equals the coefficient of $Lf$ with lowest degree in $(z-\backslash xi)$. For each formal Frobenius series solution $f$ of $Lf=0$, $\backslash alpha$ must be a root of the indicial polynomial at $\backslash xi$, i. e., $\backslash alpha$ needs to solve the indicial equation $P\_\{\backslash xi\}(\backslash alpha)\; =\; 0$.{{Cite book|title=Ordinary Differential Equations|last1=Tenenbaum|first1=Morris|last2=Pollard|first2=Harry|publisher=Dover Publications|year=1963|isbn=9780486649405|location=New York, USA|pages=Lesson 40|url=https://archive.org/details/ordinarydifferen00tene_0}}

If $\backslash xi$ is an ordinary point, the resulting indicial equation is given by $\backslash alpha^\{\backslash underline\{n\}\}=0$. If $\backslash xi$ is a regular singularity, then $\backslash deg(P\_\{\backslash xi\}(\backslash alpha))=n$ and if $\backslash xi$ is an irregular singularity,

The differential operator of order $2$, $Lf\; :=\; f\text{'}\text{'}+\backslash frac\{1\}\{z\}f\text{'}+\backslash frac\{1\}\{z^2\}f$, has a regular singularity at $z=0$. Consider a Frobenius series solution relative to $0$, $f\; :=\; z^\backslash alpha(c\_0\; +\; c\_1z\; +\; c\_2\; z^2\; +\; \backslash cdots)$ with $c\_0\backslash neq0$.

: $$

\begin{align}

Lf & = z^{\alpha-2}(\alpha(\alpha-1)c_0 + \cdots) + \frac{1}{z}z^{\alpha-1}(\alpha c_0 + \cdots) + \frac{1}{z^2}z^{\alpha}(c_0 + \cdots) \\[5pt]

& = z^{\alpha-2}c_0(\alpha(\alpha-1) + \alpha + 1) + \cdots.

\end{align}

This implies that the degree of the indicial polynomial relative to $0$ is equal to the order of the differential equation, $\backslash deg(P\_0(\backslash alpha))\; =\; \backslash deg(\backslash alpha^2\; +\; 1)\; =\; 2$.

The differential operator of order $2$, $Lf:=f\text{'}\text{'}+\backslash frac\{1\}\{z^2\}f\text{'}\; +\; f$, has an irregular singularity at $z=0$. Let $f$ be a Frobenius series solution relative to $0$.

: $$

\begin{align}

Lf & = z^{\alpha-2}(\alpha(\alpha-1)c_0 + \cdots) + \frac{1}{z^2}z^{\alpha-1}(\alpha c_0 + (\alpha+1)c_1 z + \cdots) + z^{\alpha}(c_0 + \cdots) \\[5pt]

& = z^{\alpha-3} c_0 \alpha + z^{\alpha-2}(c_0\alpha(\alpha-1) + c_1(\alpha+1)) + \cdots.

\end{align}

Certainly, at least one coefficient of the lower derivatives pushes the exponent of $z$ down. Inevitably, the coefficient of a lower derivative is of smallest exponent. The degree of the indicial polynomial relative to $0$ is less than the order of the differential equation, .

We have given a homogeneous linear differential equation $Lf=0$ of order $n$ with coefficients that are expandable as Laurent series with finite principle part. The goal is to obtain a fundamental set of formal Frobenius series solutions relative to any point $\backslash xi\backslash in\backslash mathbb\{C\}$. This can be done by the Frobenius series method, which says: The starting exponents are given by the solutions of the indicial equation and the coefficients describe a polynomial recursion. W.l.o.g., assume $\backslash xi=0$.

If $0$ is an ordinary point, a fundamental system is formed by the $n$ linearly independent formal Frobenius series solutions $\backslash psi\_1,\; z\backslash psi\_2,\; \backslash dots,\; z^\{n-1\}\backslash psi\_\{n\}$, where $\backslash psi\_i\backslash in\backslash mathbb\{C\}z$ denotes a formal power series in $z$ with $\backslash psi(0)\backslash neq0$, for $i\backslash in\backslash \{1,\backslash dots,n\backslash \}$. Due to the reason that the starting exponents are integers, the Frobenius series are power series.

If $0$ is a regular singularity, one has to pay attention to roots of the indicial polynomial that differ by integers. In this case the recursive calculation of the Frobenius series' coefficients stops for some roots and the Frobenius series method does not give an $n$-dimensional solution space. The following can be shown independent of the distance between roots of the indicial polynomial: Let $\backslash alpha\backslash in\backslash mathbb\{C\}$ be a $\backslash mu$-fold root of the indicial polynomial relative to $0$. Then the part of the fundamental system corresponding to $\backslash alpha$ is given by the $\backslash mu$ linearly independent formal solutions

: $\backslash begin\{align\}$

& z^\alpha \psi_0 \\

& z^\alpha \psi_1 + z^\alpha\log(z)\psi_0\\

& z^\alpha \psi_2 + 2z^\alpha\log(z)\psi_1 + z^\alpha\log^2(z)\psi_0\\

& \qquad \vdots\\

& z^\alpha \psi_{\mu-1} + \cdots + \binom{\mu-1}{k} z^{\alpha}\log^k(z)\psi_{\mu-k} + \cdots + z^\alpha \log^{\mu-1}(z)\psi_0

\end{align}

where $\backslash psi\_i\backslash in\backslash mathbb\{C\}z$ denotes a formal power series in $z$ with $\backslash psi(0)\backslash neq0$, for $i\backslash in\backslash \{0,\backslash dots,\backslash mu-1\backslash \}$. One obtains a fundamental set of $n$ linearly independent formal solutions, because the indicial polynomial relative to a regular singularity is of degree $n$.{{Cite book|title=Ordinary Differential Equations|last=Ince|first=Edward Lindsay|publisher=Dover Publications|year=1956|isbn=9780486158211|location=New York, USA|pages=Section 16.3}}

One can show that a linear differential equation of order $n$ always has $n$ linearly independent solutions of the form

: $\backslash exp(u(z^\{-1/s\}))\backslash cdot\; z^\backslash alpha(\backslash psi\_0(z^\{1/s\})\; +\; \backslash cdots\; +\; \backslash log^k(z)\; \backslash psi\_k(z^\{1/s\})\; +\; \backslash cdots\; +\; \backslash log^\{w\}(z)\; \backslash psi\_w(z^\{1/s\}))$

where $s\backslash in\backslash mathbb\{N\}\backslash setminus\backslash \{0\backslash \},\; u(z)\backslash in\backslash mathbb\{C\}[z]$ and $u(0)=0,\; \backslash alpha\backslash in\backslash mathbb\{C\},\; w\backslash in\backslash mathbb\{N\}$, and the formal power series $\backslash psi\_0(z),\backslash dots,\backslash psi\_w\backslash in\backslash mathbb\{C\}z$.{{Cite book|title=The Concrete Tetrahedron|last1=Kauers|first1=Manuel|last2=Paule|first2=Peter|publisher=Springer-Verlag|year=2011|isbn=9783709104453|location=Vienna, Austria|pages=Theorem 7.3}}

$0$ is an irregular singularity if and only if there is a solution with $u\backslash neq\; 0$. Hence, a differential equation is of Fuchsian type if and only if for all $\backslash xi\backslash in\backslash mathbb\{C\}\backslash cup\backslash \{\backslash infty\backslash \}$ there exists a fundamental system of Frobenius series solutions with $u=0$ at $\backslash xi$.

{{Reflist}}

• {{Cite book|title=Ordinary Differential Equations|last=Ince|first=Edward Lindsay|publisher=Dover Publications|year=1956|isbn=9780486158211|location=New York, USA}}
• {{Cite book| publisher = Clarendon Press| last = Poole| first = Edgar Girard Croker| title = Introduction to the theory of linear differential equations| location = New York| date = 1936}}
• {{Cite book|title=Ordinary Differential Equations|last1=Tenenbaum|last2=Pollard|first1=Morris|first2=Harry|publisher=Dover Publications|year=1963|isbn=9780486649405|location=New York, USA|pages=Lecture 40|url=https://archive.org/details/ordinarydifferen00tene_0}}
• {{Cite book|title=Gewöhnliche Differentialgleichungen beliebiger Ordnung|last=Horn|first=Jakob|publisher=G. J. Göschensche Verlagshandlung|year=1905|isbn=|location=Leipzig, Germany}}
• {{Cite book|title=Handbuch der Theorie der linearen Differentialgleichungen (2. Band, 1. Teil)|last=Schlesinger|first=Ludwig Lindsay|publisher=B. G.Teubner|year=1897|isbn=|location=Leipzig, Germany|pages=241 ff}}
• {{Cite book|title=Higher Special Functions|last=Lay|first=Wolfgang|publisher=Cambridge University Press|year=2024|isbn=9781009128414|location=Stuttgart, Germany|pages=114-156}}

Category:Differential equations