In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of

orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the cl ...

introduced by in the course of his work on the

Rogers–Ramanujan identities In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by , and were subsequently rediscovered (without a proof) by Sriniva ...

. They are ''q''-analogs of

ultraspherical polynomials In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval minus;1,1with respect to the weight function (1 − ''x''2)''α''–1/2. They generalize Legendre polynom ...

, and are the

Macdonald polynomials In mathematics, Macdonald polynomials ''P''λ(''x''; ''t'',''q'') are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalization in 1995. Macdonald origin ...

for the special case of the ''A''₁

affine root system In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g to ...

. and 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 In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called ...

by :

C_n(x;\beta, q) = \frace^ _2\phi_1(q^,\beta;\beta^q^;q,q\beta^e^)

where ''x'' = cos(''θ'').

References

* * * * * *{{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 Orthogonal polynomials Q-analogs