science

Meixner polynomials

Meixner polynomials is a science 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 Meixner polynomials rather than just read about it. In short: In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner (1934). They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by M n ( x , β , γ ) = ∑ k = 0 n ( − 1 ) k ( n k ) ( x k ) k !

Key takeaways

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

Reference excerpt

In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner (1934). They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by

M n ( x , β , γ ) = ∑ k = 0 n ( − 1 ) k ( n k ) ( x k ) k ! ( x + β ) n − k γ − k {\displaystyle M_{n}(x,\beta ,\gamma )=\sum _{k=0}^{n}(-1)^{k}{n \choose k}{x \choose k}k!(x+\beta )_{n-k}\gamma ^{-k}}

See also Kravchuk polynomials

References Meixner, J. (1934). "Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion". Journal of the London Mathematical Society. s1-9: 6–13. doi:10.1112/jlms/s1-9.1.6. Al-Salam, W. A. (1966). "On a characterization of Meixner's Polynomials". Quart. J. Math. 17 (1): 7–10. Bibcode:1966QJMat..17....7A. doi:10.1093/qmath/17.1.7. Atakishiyev, N. M.; Suslov, S. K. (1985). "The Hahn and Meixner polynomials of an imaginary argument and some of their applications". J. Phys. A: Math. Gen. 18 (10): 1583. Bibcode:1985JPhA...18.1583A. doi:10.1088/0305-4470/18/10/014. Andrews, George E.; Askey, Richard (1985). "Classical orthogonal polynomials". Orthogonal polynomials and applications (Bar-le-Duc, 1984). Lecture Notes in Mathematics. Vol. 1171. Berlin: Springer. pp. 36–62. doi:10.1007/BFb0076530. ISBN 978-3-540-16059-5. MR 0838970. Tratnik, M. V. (1989). "Multivariable Meixer, Krawtchouk, and Meixner-Pollaczek polynomials". J. Math. Phys. 30 (12): 2740–2749. Bibcode:1989JMP....30.2740T. doi:10.1063/1.528507. Tratnik, M. V. (1991). "Some multivariable orthogonal polynomials of the Askey tableau-discrete families". J. Math. Phys. 32 (9): 2337–2342. Bibcode:1991JMP....32.2337T. doi:10.1063/1.529158. Bavinck, H.; Vanhaeringen, H. (1994). "Difference equations for generalized Meixner Polynomials". J. Math. Anal. Appl. 184 (3): 453–463. doi:10.1006/jmaa.1994.1214. Jin, X.-S.; Wong, R. (1998). "Uniform asymptotic expansion for Meixner polynomials". Construct. Approx. 14 (1): 113–150. doi:10.1007/s003659900066. Álvarez de Morales, Maria; Pérez, T. E.; Piñar, M. A.; Ronveaux, A. (1999). "Non-standard orthogonality for Meixner Polynomials" (PDF). Electron. Trans. Numer. Anal. 9: 1–25. Archived from the original (PDF) on 2004-09-23. Retrieved 2013-03-10. Jin, X.-S.; Wong, R. (1999). "Asymptotic formulas for the zeros of Meixner Polynomials". J. Approx. Theory. 96 (2): 281–300. doi:10.1006/jath.1998.3235. Borodin, Alexei; Olshanski, Grigori (2006). "Meixner polynomials and random partitions". arXiv:math/0609806. Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Hahn Class: Definitions", 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. Boelen, L.; Filipuk, Galina; Van Assche, Walter (2011). "Recurrence coefficients of generalized Meixner polynomials and Peinlevé equations". J. Phys. A: Math. Theor. 44 (3) 035202. Bibcode:2011JPhA...44c5202B. doi:10.1088/1751-8113/44/3/035202. Wang, Xiang-Sheng; Wong, Roderick (2011). "Global asymptotics of the Meixner polynomials". Asymptot. Anal. 75 (3–4): 211–231. arXiv:1101.4370. doi:10.3233/ASY-2011-1060.

Worked examples

Example 1 — a first encounter with Meixner polynomials

Start with the simplest possible case. Write down what Meixner polynomials claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Meixner 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 Meixner 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 Meixner polynomials

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

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

Frequently asked questions

What is Meixner polynomials in simple terms?

Why does Meixner polynomials matter?

Because it connects several science 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 Meixner 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 Meixner polynomials.