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In mathematics, the '''Askey–Wilson polynomials''' (or '''''q''-Wilson polynomials''') are a family of [[orthogonal polynomials]] introduced by {{harvs|txt|last=Askey|author1-link=Richard Askey|last2=Wilson|author2-link=James A. Wilson|year=1985}} as [[q-analog]]s of the [[Wilson polynomials]]. They include many of the other orthogonal polynomials in 1 variable as [[Special case|special]] or [[limiting case]]s, described in the [[Askey scheme]]. Askey–Wilson polynomials are the special case of [[Macdonald polynomials]] (or [[Koornwinder polynomials]]) for the non-reduced [[affine root system]] of type (''C''{{su|b=1|p=&or;}}, ''C''<sub>1</sub>), and their 4 parameters ''a'', ''b'', ''c'', ''d'' correspond to the 4 orbits of roots of this root system.


They are defined by
:<math>p_n(x;a,b,c,d|q) =
(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix}
q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\
ab&ac&ad \end{matrix}
; q,q \right] </math>
where &phi; is a [[basic hypergeometric function]] and ''x'' = cos(&theta;) and (,,,)<sub>''n''</sub> is the [[q-Pochhammer symbol]]. '''Askey–Wilson functions''' are a generalization to non-integral values of ''n''.


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Askey-Wilson polynomials are the special case of [[Koornwinder polynomials]] (or [[Macdonald polynomials]]) for the non-reduced root system of type (''C''{{su|b=1|p=&or;}}, ''C''<sub>1</sub>).
 
==See also==
*[[Askey scheme]]
 
==References==
*{{Citation | authorlink=Richard Askey | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials | isbn=978-0-8218-2321-7 | mr=783216 | year=1985 | journal=Memoirs of the American Mathematical Society | issn=0065-9266 | volume=54 | issue=319 | pages=iv+55|url=http://books.google.com/books?id=9q9o03nD_xsC}}
*{{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 | mr=2128719 | year=2004 | volume=96}}
*{{dlmf|id=18.28|title=Askey-Wilson class|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
*{{Citation | first=Tom H. | last=Koornwinder | title=[http://www.scholarpedia.org/article/Askey-Wilson_polynomial Askey-Wilson polynomial] | journal=Scholarpedia | volume=7 | year=2012 | issue=7 | pages=7761}}
 
{{DEFAULTSORT:Askey-Wilson polynomials}}
[[Category:Q-analogs]]
[[Category:Hypergeometric functions]]
[[Category:Orthogonal polynomials]]

Revision as of 09:49, 29 September 2013

In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Template:Harvs as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. Askey–Wilson polynomials are the special case of Macdonald polynomials (or Koornwinder polynomials) for the non-reduced affine root system of type (CTemplate:Su, C1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system.

They are defined by

pn(x;a,b,c,d|q)=(ab,ac,ad;q)nan4ϕ3[qnabcdqn1aeiθaeiθabacad;q,q]

where φ is a basic hypergeometric function and x = cos(θ) and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.

Askey-Wilson polynomials are the special case of Koornwinder polynomials (or Macdonald polynomials) for the non-reduced root system of type (CTemplate:Su, C1).

See also

References

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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

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