en>Headbomb: Various citation cleanup (identifiers mostly) using AWB

2011-09-05T07:04:33Z

Various citation cleanup (identifiers mostly) using AWB

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{{DISPLAYTITLE: Continuous ''q''-Hermite polynomials}}
In mathematics, the '''continuous ''q''-Hermite polynomials''' are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]]. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=14}} give a detailed list of their properties.

==Definition==

The polynomials are given in terms of [[basic hypergeometric function]]s and the [[Pochhammer symbol]] by
:<math>\displaystyle </math>

==Orthogonality==
{{Empty section|date=September 2011}}

==Recurrence and difference relations==
<math>2x H_n(x|q) = H_{n+1} (x|q) + (1-q^n) H_{n-1} (x|q)</math>
with the initial conditions
<math>\textstyle H_{0} (x|q) =1, H_{-1} (x|q) = 0</math>

From the above, one can easily calculate:
{|
|-
| <math>H_{0} (x|q) =1 </math>
|-
| <math>H_{1} (x|q) = 2x</math>
|-
| <math>H_{2} (x|q) =4x^2 - (1-q^n)</math>
|-
| <math>H_{3} (x|q) =8x^3 - 4x(1-q^n)</math>
|-
| <math>H_{4} (x|q) =16x^4 - 12x^2(1-q^n) + (1-q^n)^2</math>
|-
| <math>H_{5} (x|q) =32x^5 - 32x^3(1-q^n) +6x(1-q^n)^2</math>
|}

==Rodrigues formula==
{{Empty section|date=September 2011}}

==Generating function==
:<math>\displaystyle \sum_{n=0}^{\infty} H_n(x |q) \frac{t^n}{(q;q)_n} = \frac{1}
{\left( t e^{i \theta},t e^{-i \theta};q \right)_{\infty}} </math>
where <math>\textstyle x=\cos \theta</math>

==Relation to other polynomials==
{{Empty section|date=September 2011}}

==References==

*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
*{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
*{{dlmf|id=18|title=|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}

[[Category:Orthogonal polynomials]]
[[Category:Q-analogs]]
[[Category:Special hypergeometric functions]]

Dual q-Krawtchouk polynomials - Revision history

en>Headbomb: Various citation cleanup (identifiers mostly) using AWB