In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by , who was inspired by the continuous ''q''-Hermite polynomials studied by

Leonard James Rogers Leonard James Rogers Royal Society, FRS (30 March 1862 – 12 September 1933) was a British mathematician who was the first to discover the Rogers–Ramanujan identity and Hölder's inequality, and who introduced Rogers polynomials. The Roger ...

. They are given by :

h_n(x;q) = \sum_^n\fracx^k

where (''q'';''q'')_''n'' is the descending

q-Pochhammer symbol In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symb ...

. Furthermore, the

h_n(x;q)

satisfy (for

n \ge 1

) the recurrence relation :

h_(x;q) = (1+x)h_n(x;q) + x(q^n-1)h_(x;q)

with

h_0(x;q)=1

and

h_1(x;q)=1+x

References

* * {{DEFAULTSORT:Rogers-Szego polynomials Orthogonal polynomials Q-analogs