Table of Contents
1. Holonomic Sequence
- P-recursive Sequence, D-finite Sequence
- The coefficients are functions of \(n\).
2. Holonomic Function
- Differentiably Finite Function, D-finite Function
2.1. Definition
- Smooth Function of several variables that is a solution of a system of linear homogeneous differential equation with polynomial coefficients, and satisfies suitable dimension condition in terms of D-modules theory.
- It is an element of a holonomic module of smooth functions.
- For a field \(\mathbb{K}\) with ../algebra/Ring Theory.html#org65f1e63 0,
- A function \(f\) is called holonomic, if there exits nonzero
polynomials \((a_i)_{i=0}^r \subset \mathbb{K}[x]\), such that
- \[ \sum_{i=0}^r a_if^{(i)} = 0. \]
2.1.1. Annihilating Operator
- \[ A := \sum_{i=1}^r a_iD^i \]
- is the annihilating operator of \(f\).
- And it generates an ideal in the ring \(\mathbb{K}[x][D]\), called the annihilator of \(f\).
- \(r\) is called the order of the annihilating operator, and by extension the function \(f\) is of order \(r\) if annihilating operator of such order exists.
3. Properties
3.1. Finite Representation
- It can be represented by a finite amount of data, namely an annihilating operator and a finite set of initial values.
3.2. Closure
- Holonomic functions are closed under
- Linear combination
- Point-wise product
- Hadamard product
- Integration
- …