Continuous q-Hermite polynomials
- Continuous q-Hermite polynomials
-
In mathematics, the continuous q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
Orthogonality
Recurrence and difference relations
Rodrigues formula
Generating function
Relation to other polynomials
References
- Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR2128719
- 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, MR2656096
- Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/18
Wikimedia Foundation.
2010.
Look at other dictionaries:
Hermite polynomials — In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical… … Wikipedia
Continuous big q-Hermite polynomials — In mathematics, the continuous big q Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their… … Wikipedia
Charles Hermite — Hermite redirects here. For other uses, see Hermite (disambiguation). Charles Hermite Charles Hermite circa 1901 … Wikipedia
Classical orthogonal polynomials — In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical… … Wikipedia
Orthogonal polynomials — In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the… … Wikipedia
Chebyshev polynomials — Not to be confused with discrete Chebyshev polynomials. In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of orthogonal polynomials which are related to de Moivre s formula and which can be defined… … Wikipedia
Cubic Hermite spline — In the mathematical subfield of numerical analysis a cubic Hermite spline (also called cspline), named in honor of Charles Hermite, is a third degree spline with each polynomial of the spline in Hermite form. The Hermite form consists of two… … Wikipedia
Fourier transform — Fourier transforms Continuous Fourier transform Fourier series Discrete Fourier transform Discrete time Fourier transform Related transforms The Fourier transform is a mathematical operation that decomposes a function into its constituent… … Wikipedia
Weierstrass transform — In mathematics, the Weierstrass transform [Ahmed I. Zayed, Handbook of Function and Generalized Function Transformations , Chapter 18. CRC Press, 1996.] of a function f : R rarr; R is the function F defined by:F(x)=frac{1}{sqrt{4piint {… … Wikipedia
Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function … Wikipedia
