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In mathematics, the '''little ''q''-Laguerre polynomials''' ''p''<sub>''n''</sub>(''x'';''a''|''q'') or  '''Wall polynomials''' ''W''<sub>''n''</sub>(''x''; ''b'',''q'') are a family of basic hypergeometric [[orthogonal polynomials]] in the basic [[Askey scheme]] closely related to a continued fraction studied by {{harvs|txt|last=Wall|authorlink=Hubert Stanley Wall|year=1941}}. (The term "Wall polynomial" is also used for an unrelated [[Wall polynomial]] in the theory of classical groups.)
{{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  p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)  </math>
 
==Orthogonality==
{{Empty section|date=September 2011}}
 
==Recurrence and difference relations==
{{Empty section|date=September 2011}}
 
==Rodrigues formula==
{{Empty section|date=September 2011}}
 
==Generating function==
{{Empty section|date=September 2011}}
 
==Relation to other polynomials==
{{Empty section|date=September 2011}}
 
==References==
 
*{{Citation | last1=Chihara | first1=Theodore Seio | title=An introduction to orthogonal polynomials | url=http://books.google.com/books?id=IkCJSQAACAAJ | publisher=Gordon and Breach Science Publishers | location=New York | series=Mathematics and its Applications | isbn=978-0-677-04150-6 | mr=0481884 |id= Reprinted by Dover 2011, ISBN 978-0-486-47929-3 | year=1978 | volume=13}}
*{{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}}
*{{Citation | last1=Van Assche | first1=Walter | last2=Koornwinder | first2=Tom H. | title=Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials | doi=10.1137/0522019 | mr=MR1080161 | year=1991 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=22 | issue=1 | pages=302–311}}
*{{Citation | last1=Wall | first1=H. S. | title=A continued fraction related to some partition formulas of Euler | jstor=2303599 | mr=0003641 | year=1941 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=48 | pages=102–108}}
 
[[Category:Orthogonal polynomials]]
[[Category:Q-analogs]]
[[Category:Special hypergeometric functions]]

Revision as of 21:29, 26 January 2014

In mathematics, the little q-Laguerre polynomials pn(x;a|q) or Wall polynomials Wn(x; b,q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to a continued fraction studied by Template:Harvs. (The term "Wall polynomial" is also used for an unrelated Wall polynomial in the theory of classical groups.) Template:Harvs give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

pn(x;a|q)=2ϕ1(q−n,0;aq;q,qx)=1(a−1q−n;q)n2ϕ0(q−n,x−1;;q,x/a)

Orthogonality

Template:Empty section

Recurrence and difference relations

Template:Empty section

Rodrigues formula

Template:Empty section

Generating function

Template:Empty section

Relation to other polynomials

Template:Empty section

References

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    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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  • Template:Dlmf
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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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.

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    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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