mathematics

Lommel polynomial

Lommel polynomial is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lommel polynomial rather than just read about it. In short: A Lommel polynomial Rm,ν(z) is a polynomial in 1/z giving the recurrence relation J m + ν ( z ) = J ν ( z ) R m , ν ( z ) − J ν − 1 ( z ) R m − 1 , ν + 1 ( z ) {\displaystyle \displaystyle J_{m+\nu }(z)=J_{\nu }(z)R_{m,\nu }(z)-J_{\nu -1}(z)R_{m-1,\nu +1}(z)} where Jν(z) is a Bessel function of the first kind. They are given explicitly by R m , ν ( z ) = ∑ n = 0 [ m / 2 ] ( − 1 ) n ( m − n ) ! Γ ( ν + m − n ) n !

Key takeaways

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

Reference excerpt

A Lommel polynomial Rm,ν(z) is a polynomial in 1/z giving the recurrence relation

J m + ν ( z ) = J ν ( z ) R m , ν ( z ) − J ν − 1 ( z ) R m − 1 , ν + 1 ( z ) {\displaystyle \displaystyle J_{m+\nu }(z)=J_{\nu }(z)R_{m,\nu }(z)-J_{\nu -1}(z)R_{m-1,\nu +1}(z)}

where Jν(z) is a Bessel function of the first kind. They are given explicitly by

R m , ν ( z ) = ∑ n = 0 [ m / 2 ] ( − 1 ) n ( m − n ) ! Γ ( ν + m − n ) n ! ( m − 2 n ) ! Γ ( ν + n ) ( z / 2 ) 2 n − m . {\displaystyle R_{m,\nu }(z)=\sum _{n=0}^{[m/2]}{\frac {(-1)^{n}(m-n)!\Gamma (\nu +m-n)}{n!(m-2n)!\Gamma (\nu +n)}}(z/2)^{2n-m}.}

See also Lommel function Neumann polynomial

References

Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756 Ivanov, A. B. (2001) [1994], "Lommel polynomial", Encyclopedia of Mathematics, EMS Press

Worked examples

Example 1 — a first encounter with Lommel polynomial

Start with the simplest possible case. Write down what Lommel polynomial 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 Lommel polynomial 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 Lommel polynomial 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 Lommel polynomial

In research: Lommel polynomial 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 Lommel polynomial 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: Lommel polynomial is common in secondary-school and first-year university syllabi. It links to neighbouring topics Polynomial stubs, Polynomials, Special functions, so understanding it makes those chapters shorter.
In everyday life: Look for Lommel polynomial 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.

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How to study Lommel polynomial in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lommel polynomial 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 Lommel polynomial out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lommel polynomial in simple terms?

A Lommel polynomial Rm,ν(z) is a polynomial in 1/z giving the recurrence relation J m + ν ( z ) = J ν ( z ) R m , ν ( z ) − J ν − 1 ( z ) R m − 1 , ν + 1 ( z ) {\displaystyle \displaystyle J_{m+\nu }(z)=J_{\nu }(z)R_{m,\nu }(z)-J_{\nu -1}(z)R_{m-1,\nu +1}(z)} where Jν(z) is a Bessel function of the…

Why does Lommel polynomial 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 Lommel polynomial?

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 Lommel polynomial.