“Al-Salam–Chihara polynomials”的意思、由来-开放百科全书

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Al-Salam–Chihara polynomials

释义

Definition
Orthogonality
Recurrence and difference relations
Rodrigues formula
Generating function
Relation to other polynomials
References

In mathematics, the Al-Salam–Chihara polynomials Q_n(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by {{harvs|txt|last=Al-Salam|authorlink=Waleed Al-Salam|last2=Chihara|author2-link=Theodore Seio Chihara|year=1976}}. {{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=14.8}} give a detailed list of the properties of Al-Salam–Chihara polynomials.

Definition

The Al-Salam–Chihara polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

where x = cos(θ).

Orthogonality

Recurrence and difference relations

Rodrigues formula

Generating function

Relation to other polynomials

References

{{Citation | last1=Al-Salam | first1=W. A. | last2=Chihara | first2=Theodore Seio | title=Convolutions of orthonormal polynomials | doi=10.1137/0507003 | mr=0399537 | year=1976 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=7 | issue=1 | pages=16–28}}
{{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=18.28iii|title=Al-Salam–Chihara polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}

3 : Orthogonal polynomials|Q-analogs|Special hypergeometric functions

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