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= Fuchsian Theory = At any ordinary point of a homogeneous linear differential equation of order $$n$$ there exists a fundamental system of $$n$$ linearly independent classical solutions. At singularities the dimension of the solution space can stay below the order of the differential equation, depending on the function space in which solutions are searched. For example, when searching solutions in the set of rational functions, power series or Laurent series the amount of linearly independent solutions stays below $$n$$ at singularities. The Fuchsian theory of linear differential equations, which is named after Lazarus Immanuel Fuchs, enables the investigation of the solution’s behavior at singularities. In the Fuchsian Theory only formal solutions are considered.

Generalized Series Solutions
The generalized series at $$\xi\in\mathbb{C}$$ is defined by

$$(z-\xi)^{\alpha}\sum_{k=0}^{\infty}c_k(z-\xi)^k, \text{ with } \alpha,c_k \in\mathbb{C} \text{ and }c_0\neq0$$,

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

$${d^nf \over d z^n} = (z-\xi)^{\alpha-n}\sum_{k=0}^{\infty}(\alpha+k)^{\underline{n}}c_k(z-\xi)^k$$,

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

Indicial Equation
Let $f:=(z-\xi)^{\alpha}\sum_{k=0}^{\infty}c_k(z-\xi)^k$ be a Frobenius series relative to $$\xi \in \mathbb{C}$$. Let $$L $$ be a homogeneous linear differential operator of order $$n$$. $$Lf$$ is a Frobenius series if and only if all coefficients are expandable as Laurent series with finite principle part at $$\xi$$. Assume that $$Lf$$ is a Frobenius series, then it is of the form $$Lf=(z-\xi)^{\beta}\psi(z)$$, with a certain $$\beta\in\mathbb{C}$$ and a certain power series $$\psi(z)$$ in $$(z-\xi)$$. The indicial polynomial is defined by $$P_{\xi}:=\psi(0)$$, i. e.,  $$P_{\xi}$$ equals the coefficient of $$Lf$$ with lowest degree in $$(z-\xi)$$ which is a polynomial in $$\alpha$$. $$f$$ is a formal solution of $$Lf=0$$ if and only if $$\alpha$$ is a root of the indicial polynomial, i. e., $$\alpha$$ needs to solve the indicial equation $$P_{\xi}(\alpha) = 0$$. The solutions of the indicial equation relative to $$\xi$$ gives the starting exponents of the formal Frobenius series solutions at $$\xi$$.

If $$\xi$$ is an ordinary point, the resulting indicial equation is given by $$\alpha^{\underline{n}}=0$$. If $$\xi$$ is a regular singularity, then $$\deg(P_{\xi}(\alpha))=n$$ and if $$\xi$$ is an irregular singularity, $$\deg(P_{\xi}(\alpha))<n$$ holds. This is illustrated by the later examples. The indicial equation relative to $$\xi=\infty$$ is defined by the inidical equation of $$\widetilde{L}f$$, where $$\widetilde{L}$$ denotes the differential operator $$L$$ transformed by $$x=z^{-1}$$ and $$f$$ a Frobenius series in $$x$$.

Example: Regular Singularity
The differential operator of order $$2$$, $$Lf := f''+\frac{1}{z}f'+\frac{1}{z^2}f$$, has a regular singularity at $$z=0$$. Consider a Frobenius series solution relative to $$0$$, $$f := z^{\alpha}(c_0 + c_1z + c_2 z^2 + \cdots)$$ with $$c_0\neq0$$.

$$Lf = z^{\alpha-2}(\alpha(\alpha-1)c_0 + \dots) + \frac{1}{z}z^{\alpha-1}(\alpha c_0 + \dots) + \frac{1}{z^2}z^{\alpha}(c_0 + \dots) = z^{\alpha-2}c_0(\alpha(\alpha-1) + \alpha + 1) + \dots$$.

This implies that the degree of the indicial polynomial relative to $$0$$ is equal to the order of the differential equation, $$\deg(P_0(\alpha)) = \deg(\alpha^2 + 1) = 2$$.

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

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

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, $$\deg(P_0(\alpha)) = \deg(\alpha) = 1 < 2$$.

Formal Fundamental Systems
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, i. e., a Frobenius series $$f$$ makes the ansatz $$Lf$$ a Frobenius series. The goal is to obtain a fundamental set of formal Frobenius series solutions relative to any point $$\xi\in\mathbb{C}$$. 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 $$\xi=0$$.

Fundamental System at Ordinary Point
If $$0$$ is an ordinary point, a fundamental system is formed by the $$n$$ linearly independent formal Frobenius series solutions $$\psi_1, z\psi_2, \dots, z^{n-1}\psi_{n}$$, where $\psi_i\in\mathbb{C}z$ denotes a formal power series in $$z$$ with $$\psi(0)\neq0$$, for $$i\in\{1,\dots,n\}$$. Due to the reason that the starting exponents are integers, the Frobenius series are power series.

Fundamental System at Regular Singularity
If $$0$$ is a regular singularity, one have 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 $$\alpha\in\mathbb{C}$$ be a $$\mu$$-fold root of the indicial polynomial relative to $$0$$. Then the part of the fundamental system corresponding to $$\alpha$$ is given by the $$\mu$$ linearly independent formal solutions

$$\begin{align} z^{\alpha}\psi_0\\ z^{\alpha}\psi_1 + z^{\alpha}\log(z)\psi_0\\ z^{\alpha}\psi_2 + 2z^{\alpha}\log(z)\psi_1 + z^{\alpha}\log^2(z)\psi_0\\ \dots\\ z^{\alpha}\psi_{\mu-1} + \cdots + \binom{\mu-1}{k} z^{\alpha}\log^k(z)\psi_{\mu-k} + \cdots + z^{\alpha}\log^{\mu-1}(z)\psi_0 \end{align} $$

where $\psi_i\in\mathbb{C}z$ denotes a formal power series in $$z$$ with $$\psi(0)\neq0$$, for $$i\in\{0,\dots,\mu-1\}$$. One obtains a fundamental set of $$n$$ linearly independent formal solutions, because the indicial polynomial relative to a regular singularity is of degree $$n$$.

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

$$exp(u(z^{-\frac{1}{s}}))\cdot z^{\alpha}(\psi_0(z) + \cdots + \log^k(x) \psi_{k}(z) + \cdots + \log^{w}(x) \psi_w(z))$$

where $$s\in\mathbb{N}\setminus\{0\}, u(z)\in\mathbb{C}[z]$$ and $$u(0)=0, \alpha\in\mathbb{C}, w\in\mathbb{N}$$, and the formal power series $$\psi_0(z),\dots,\psi_w\in\mathbb{C}z$$.

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

= Fuchs Relation =

The Fuchs relation is described by 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. For Fuchsian equations a formal fundamental system consisting of Frobenius series solutions exists at any point, due to the Fuchsian Theory.

Coefficients of a Fuchsian Equation
Let $$a_1, \dots, a_r \in \mathbb{C}$$ be the $$r$$ regular singularities in the finite part of the complex plane of the linear differential equation$$Lf := \frac{d^nf}{dz^n} + q_1\frac{d^{n-1}f}{dz^{n-1}} + \cdots + q_{n-1}\frac{df}{dz} + q_nf$$

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 form

$$q_i(z) = \frac{Q_i(z)}{\psi^{i}}$$

with the polynomial $\psi := \prod_{j=0}^r (z-a_j) \in\mathbb{C}[z]$ and certain polynomials $$Q_i \in \mathbb{C}[z]$$ for $$i\in \{1,\dots,n\}$$, such that $$\deg(Q_i) \leq i(r-1)$$. This means $$a_1, \dots, a_r$$ are poles of the coefficients $$q_i$$ with multiplicity at most $$i$$.

Fuchs Relation
Let $$Lf=0$$ be a Fuchsian equation of order $$n$$ and $$a_1, \dots, a_r\in\mathbb{C}$$ be the singularities of $$Lf=0$$ in the finite part of the complex plane. Let $$\alpha_{i1},\dots,\alpha_{in}\in\mathbb{C}$$ be the roots of the indicial polynomial relative to $$a_i$$, for $$i\in\{1,\dots,r\}$$, and let $$-\beta_1,\dots,-\beta_n\in\mathbb{C}$$ be the roots of the indicial polynomial relative to $$\infty$$. Then the so called Fuchs relation holds:

$$\sum_{i=1}^r \sum_{k=1}^n \alpha_{ik} + \sum_{k=1}^n \beta_{k} = \frac{n(n-1)}{2}$$.

Let $$P_{\xi}$$ denote the indicial polynomial relative to $$\xi\in\mathbb{C}\cup\{\infty\}$$ of the Fuchsian equation $$Lf=0$$. Then the Fuchs relation is equivalent to$$\sum_{\xi\in\{a_1,\dots,a_r\}} Tr(P_{\xi}) - Tr(P_{\infty}) = \frac{n(n-1)}{2}$$

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

The Fuchs relation can be rewritten as infinite sum. Define $$defect: \mathbb{C}\cup\{\infty\}\to\mathbb{C}$$ as

$$defect(\xi):= \begin{cases} Tr(P_{\xi}) - \frac{n(n-1)}{2}\text{, for }\xi\in\mathbb{C}\\ Tr(P_{\xi}) + \frac{n(n-1)}{2}\text{, for }\xi=\infty. \end{cases}$$

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

$$\sum_{\xi\in\mathbb{C}\cup\{\infty\}} defect(\xi) = 0$$.