Rogers polynomials

In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by {{harvs|txt|authorlink=Leonard James Rogers|last=Rogers|year1=1892|year2=1893|year3=1894}} in the course of his work on the Rogers–Ramanujan identities. They are q-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the A₁ affine root system {{harv|Macdonald|2003|loc=p.156}}.

{{harvtxt|Askey|Ismail|1983}} and {{harvtxt|Gasper|Rahman|2004|loc=7.4}} discuss the properties of Rogers polynomials in detail.

Definition

The Rogers polynomials can be defined in terms of the q-Pochhammer symbol and the basic hypergeometric series by

: $C_n(x;\beta|q) = \frac{(\beta;q)_n}{(q;q)_n}e^{in\theta} {}_2\phi_1(q^{-n},\beta;\beta^{-1}q^{1-n};q,q\beta^{-1}e^{-2i\theta})$

where x = cos(θ).

References

{{Citation | last1=Askey | first1=Richard | last2=Ismail | first2=Mourad E. H. | editor1-last=Erdős | editor1-first=Paul | title=Studies in pure mathematics. To the memory of Paul Turán. | chapter-url=https://books.google.com/books?id=WePuAAAAMAAJ | publisher=Birkhäuser | location=Basel, Boston, Berlin | isbn=978-3-7643-1288-6 | mr=820210 | year=1983 | chapter=A generalization of ultraspherical polynomials | pages=55–78}}
{{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}}
{{Citation | last1=Macdonald | first1=I. G. | author1-link=Ian G. Macdonald | title=Affine Hecke algebras and orthogonal polynomials | publisher=Cambridge University Press | series=Cambridge Tracts in Mathematics | isbn=978-0-521-82472-9 | doi=10.1017/CBO9780511542824 | mr=1976581 | year=2003 | volume=157| url=http://www.numdam.org/item/SB_1994-1995__37__189_0/ }}
{{Citation | last1=Rogers | first1=L. J. | title=On the expansion of some infinite products | doi=10.1112/plms/s1-24.1.337 | jfm=25.0432.01 | year=1892 | journal=Proc. London Math. Soc. | volume=24 | issue=1 | pages=337–352 | url=https://zenodo.org/record/1433380 }}
{{Citation | last1=Rogers | first1=L. J. | title=Second Memoir on the Expansion of certain Infinite Products | doi=10.1112/plms/s1-25.1.318 | year=1893 | journal=Proc. London Math. Soc. | volume=25 | issue=1 | pages=318–343| url=https://zenodo.org/record/1447732 }}
{{Citation | last1=Rogers | first1=L. J. | title=Third Memoir on the Expansion of certain Infinite Products | doi=10.1112/plms/s1-26.1.15 | year=1894 | journal=Proc. London Math. Soc. | volume=26 | issue=1 | pages=15–32| url=https://zenodo.org/record/1447734 }}

Category:Orthogonal polynomials

Category:Q-analogs