mathematics

Stieltjes–Wigert polynomials

Stieltjes–Wigert polynomials 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 Stieltjes–Wigert polynomials rather than just read about it. In short: In mathematics, Stieltjes–Wigert polynomials (named after Thomas Jan Stieltjes and Carl Severin Wigert) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, for the weight function w ( x ) = k π x − 1 / 2 exp ⁡ ( − k 2 log 2 ⁡ x ) {\displaystyle w(x)={\frac {k}{\sqrt {\pi }}}x^{-1/2}\exp(-k^{2}\log ^{2}x)} on the positive real line x > 0. The moment problem for the Stieltjes–Wigert…

Key takeaways

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

Reference excerpt

w ( x ) = k π x − 1 / 2 exp ⁡ ( − k 2 log 2 ⁡ x ) {\displaystyle w(x)={\frac {k}{\sqrt {\pi }}}x^{-1/2}\exp(-k^{2}\log ^{2}x)}

on the positive real line x > 0. The moment problem for the Stieltjes–Wigert polynomials is indeterminate; in other words, there are many other measures giving the same family of orthogonal polynomials (see Krein's condition). Koekoek et al. (2010) give in Section 14.27 a detailed list of the properties of these polynomials.

Definition The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

S n ( x ; q ) = 1 ( q ; q ) n

1 ϕ 1 ( q − n , 0 ; q , − q n + 1 x ) , {\displaystyle \displaystyle S_{n}(x;q)={\frac {1}{(q;q)_{n}}}{}_{1}\phi _{1}(q^{-n},0;q,-q^{n+1}x),}

where

q = exp ⁡ ( − 1 2 k 2 ) . {\displaystyle q=\exp \left(-{\frac {1}{2k^{2}}}\right).}

Orthogonality Since the moment problem for these polynomials is indeterminate there are many different weight functions on [0,∞] for which they are orthogonal. Two examples of such weight functions are

1 ( − x , − q x − 1 ; q ) ∞ {\displaystyle {\frac {1}{(-x,-qx^{-1};q)_{\infty }}}}

and

k π x − 1 / 2 exp ⁡ ( − k 2 log 2 ⁡ x ) . {\displaystyle {\frac {k}{\sqrt {\pi }}}x^{-1/2}\exp \left(-k^{2}\log ^{2}x\right).}

Notes

References Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719 Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096 Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Ch. 18, Orthogonal polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. Szegő, Gábor (1975), Orthogonal Polynomials, Colloquium Publications 23, American Mathematical Society, Fourth Edition, ISBN 978-0-8218-1023-1, MR 0372517 Stieltjes, T. -J. (1894), "Recherches sur les fractions continues", Ann. Fac. Sci. Toulouse (in French), VIII (4): 1–122, doi:10.5802/afst.108, JFM 25.0326.01, MR 1344720 Wang, Xiang-Sheng; Wong, Roderick (2010). "Uniform asymptotics of some q-orthogonal polynomials". J. Math. Anal. Appl. 364 (1): 79–87. doi:10.1016/j.jmaa.2009.10.038. Wigert, S. (1923), "Sur les polynomes orthogonaux et l'approximation des fonctions continues", Arkiv för matematik, astronomi och fysik (in French), 17: 1–15, JFM 49.0296.01

Worked examples

Example 1 — a first encounter with Stieltjes–Wigert polynomials

Start with the simplest possible case. Write down what Stieltjes–Wigert polynomials 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 Stieltjes–Wigert polynomials 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 Stieltjes–Wigert polynomials 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 Stieltjes–Wigert polynomials

In research: Stieltjes–Wigert polynomials 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 Stieltjes–Wigert polynomials 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: Stieltjes–Wigert polynomials is common in secondary-school and first-year university syllabi. It links to neighbouring topics Orthogonal polynomials, Special hypergeometric functions, so understanding it makes those chapters shorter.
In everyday life: Look for Stieltjes–Wigert polynomials 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 Stieltjes–Wigert polynomials in 20 minutes

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

Frequently asked questions

What is Stieltjes–Wigert polynomials in simple terms?

In mathematics, Stieltjes–Wigert polynomials (named after Thomas Jan Stieltjes and Carl Severin Wigert) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, for the weight function w ( x ) = k π x − 1 / 2 exp ⁡ ( − k 2 log 2 ⁡ x ) {\displaystyle w(x)={\frac {k}{\sq…

Why does Stieltjes–Wigert polynomials 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 Stieltjes–Wigert polynomials?

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 Stieltjes–Wigert polynomials.