Fuchsian theory

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 differential equations. 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 Lf=f^{(n)} + q_1f^{(n-1)} + \cdots + q_nf be a linear differential operator of order n with one valued coefficient functions q_1, \dots, q_n. Let all coefficients q_1,\dots,q_n be expandable as Laurent series with finite principle part at \xi. Then there exists a smallest N\in\mathbb{N} such that (z-\xi)^Nq_i is a power series for all i\in\{1,\dots, n\}. Hence, Lf is a Frobenius series of the form Lf=(z-\xi)^{\alpha-n-N}\psi(z), with a certain power series \psi(z) in (z-\xi). The indicial polynomial is defined by P_{\xi}:=\psi(0) which is a polynomial in \alpha, i.e., P_{\xi} equals the coefficient of Lf with lowest degree in (z-\xi). For each formal Frobenius series solution f of Lf=0, \alpha must be a root of the indicial polynomial at \xi, i. e., \alpha needs to solve the indicial equation P_{\xi}(\alpha) = 0. 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)) holds. This is illustrated by the later examples. The indicial equation relative to \xi=\infty is defined by the indicial equation of \widetilde{L}f, where \widetilde{L} denotes the differential operator L transformed by z=x^{-1}which is a linear differential operator in x, at x=0. 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. : \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, \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. : \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, \deg(P_0(\alpha)) = \deg(\alpha) = 1 . == 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. 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. 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^{-1/s}))\cdot z^\alpha(\psi_0(z^{1/s}) + \cdots + \log^k(z) \psi_k(z^{1/s}) + \cdots + \log^{w}(z) \psi_w(z^{1/s})) 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. == References ==