직교다항식

개요

• 직교다항식(orthogonal polynomials)
  • 직교성과 완비성
  • 3항 점화식 (3-term recurrence relation) 연분수와 관계
  • 삼각함수 곱셈공식의 일반화 linearization of products
  • 스텀-리우빌 문제

예

• 자코비 다항식
• 체비셰프 다항식
• 르장드르 다항식
• 라게르 다항식
• 게겐바워 다항식(ultraspherical polynomials)
• 에르미트 다항식(Hermite polynomials)
• 윌슨 다항식
• 로저스-세괴 다항식 (Rogers-Szegő polynomials)

메모

• Difference Equations, Continued Fractions, and Orthogonal Polynomials (Walk Into a Bar) http://math.illinoisstate.edu/schebol/algebra-seminar-files/ortho.pdf
• Why is electrostatics in the complex plane interesting from a mathematical point of view?
• http://www.maths.leeds.ac.uk/~kisilv/courses/sp-funct.pdf
• 감마함수
• Digamma 함수
• 오일러 베타적분(베타함수)

관련된 항목들

• 공대수 (coalgebra)
• 다이로그 함수(dilogarithm)
• 삼각함수에는 왜 공식이 많은가?
• 오일러(1707-1783)

메모

• Dumitriu, Ioana, Alan Edelman, and Gene Shuman. “MOPS: Multivariate Orthogonal Polynomials (symbolically).” arXiv:math-ph/0409066, September 23, 2004. http://arxiv.org/abs/math-ph/0409066.

리뷰, 에세이, 강의노트

• Wasson, Ryan D., and Robert Gilmore. 2013. “An Overview of the Relationship between Group Theory and Representation Theory to the Special Functions in Mathematical Physics.” arXiv:1309.2544 [math-Ph], September. http://arxiv.org/abs/1309.2544.
• Ehrenpreis, Leon. 2010. “Special Functions.” Inverse Problems and Imaging 4 (4): 639–47. doi:10.3934/ipi.2010.4.639.
• The History and Future of Special Functions Stephen Wolfram, 2005
• Kalnins, Special functions, Lie theory and partial differential equations, 1997
• Koekoek, Roelof, and Rene F. Swarttouw. "The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue." arXiv preprint math/9602214 (1996). http://arxiv.org/abs/math/9602214
• Kirillov, A. A., & Etingof, P. I. I. (1994). A unified representation-theoretic approach to special functions. Functional Analysis and Its Applications, 28(1), 73-76.

관련논문

• Koornwinder, Tom H. “Quadratic Transformations for Orthogonal Polynomials in One and Two Variables.” arXiv:1512.09294 [math], December 31, 2015. http://arxiv.org/abs/1512.09294.
• Odake, Satoru. “Recurrence Relations of the Multi-Indexed Orthogonal Polynomials : III.” arXiv:1509.08213 [hep-Th, Physics:math-Ph, Physics:nlin], September 28, 2015. http://arxiv.org/abs/1509.08213.
• Borzov, V. V., and E. V. Damaskinsky. ‘Comment on “On the Dimensions of the Oscillator Algebras Induced by Orthogonal Polynomials” [J. Math. Phys. {\bf 55}, 093511 (2014)]’. arXiv:1503.08202 [math-Ph], 27 March 2015. http://arxiv.org/abs/1503.08202.
• Honnouvo, G., and K. Thirulogasanthar. ‘On the Dimensions of the Oscillator Algebras Induced by Orthogonal Polynomials’. arXiv:1305.2509 [math-Ph], 11 May 2013. http://arxiv.org/abs/1305.2509.
• Dimitrov, Dimitar, and Yuan Xu. “Slater Determinants of Orthogonal Polynomials.” arXiv:1412.0326 [math-Ph], November 30, 2014. http://arxiv.org/abs/1412.0326.
• Jafarov, E. I., N. I. Stoilova, and J. Van der Jeugt. ‘On a Pair of Difference Equations for the \(_4F_3\) Type Orthogonal Polynomials and Related Exactly-Solvable Quantum Systems’. arXiv:1411.6125 [math-Ph], 22 November 2014. http://arxiv.org/abs/1411.6125.