P-recursive equation

Key takeaways

P-recursive equation belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect P-recursive equation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of P-recursive equation from memory before moving on to harder problems.

Reference excerpt

In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients. These equations play an important role in different areas of mathematics, specifically in combinatorics. The sequences which are solutions of these equations are called holonomic, P-recursive, or D-finite. From the late 1980s, the first algorithms were developed to find solutions for these equations. Sergei A. Abramov, Marko Petkovšek and Mark van Hoeij described algorithms to find polynomial, rational, hypergeometric and d'Alembertian solutions.

Definition Let F {\textstyle \mathbb {F} } be a field of characteristic zero, meaning there is no m ∈ N {\textstyle m\in \mathbb {N} } such that 0 = ∑ k = 1 m 1 {\textstyle 0=\sum _{k=1}^{m}1} , and n ∈ N {\textstyle n\in \mathbb {N} } . A sequence ( x k ) k ∈ N 0 ∈ F N 0 {\textstyle (x_{k})_{k\in \mathbb {N} _{0}}\in \mathbb {F} ^{\mathbb {N} _{0}}} is called P-recursive if it satisfies a linear recursive equation with coefficients in F [ n ] {\textstyle \mathbb {F} [n]} meaning there exists a sequence ( p k ( n ) ) k ∈ N 0 ∈ ( F [ n ] ) N 0 {\textstyle (p_{k}(n))_{k\in \mathbb {N} _{0}}\in (\mathbb {F} [n])^{\mathbb {N} _{0}}} a polynomial y ( n ) ∈ F [ n ] {\textstyle y(n)\in \mathbb {F} [n]} such that ∑ k = 0 ∞ p k ( n ) ⋅ x n + k = y ( n ) . {\displaystyle \sum \limits _{k=0}^{\infty }p_{k}(n)\cdot x_{n+k}=y(n).} Linear equations of this form are called P-recursive equations. Solutions to this equation are generally in x n ∈ F [ n ] ¯ {\textstyle x_{n}\in {\overline {\mathbb {F} [n]}}} meaning the completion of F [ n ] {\textstyle \mathbb {F} [n]} , More specifically a equation ∑ k = 0 m p k ( n ) ⋅ x n + k = y ( n ) {\displaystyle \sum \limits _{k=0}^{m}p_{k}(n)\cdot x_{n+k}=y(n)} for a m ∈ N {\textstyle m\in \mathbb {N} } is said to be of order m {\textstyle m} if and only if p 0 , p m {\textstyle p_{0},\,p_{m}} are non-zero. Some authors further more require the equation to be homogenous, meaning y {\textstyle y} to be zero, for it to be called P-recursive. Further more a power series is called D-finite if it's coefficients are P-recursive.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with P-recursive equation

Start with the simplest possible case. Write down what P-recursive equation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to P-recursive equation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about P-recursive equation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of P-recursive equation

In research: P-recursive equation appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses P-recursive equation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: P-recursive equation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Polynomials, so understanding it makes those chapters shorter.
In everyday life: Look for P-recursive equation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study P-recursive equation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what P-recursive equation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain P-recursive equation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is P-recursive equation in simple terms?

In mathematics a P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence equations (or linear recurrence relations or linear difference equations) with polynomial coefficients.

Why does P-recursive equation matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study P-recursive equation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on P-recursive equation.