Rogers–Szegő polynomials explained

Rogers–Szegő polynomials should not be confused with Rogers polynomials.

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. They are given by

h _{n(x;q)} = \sum ^{n ((q;q) _{n})/((q;q) _{k(q;q)} _{n-k})} _{k =0} x ^{k} where (q;q)_n is the descending q-Pochhammer symbol.

Furthermore, the

h _{n(x;q)} satisfy (for n \ge 1 ) the recurrence relation^[1] h _{n+1} (x;q) = ( 1+ x)h _{n(x;q)} + x(q ^{n -1 )h} _{n -1} (x;q) with h _{0(x;q)=1} and h _{1(x;q)=1+x} .

Notes and References

1. An enumeration of flags in finite vector spaces. C. Ryan. Vinroot . The Electronic Journal of Combinatorics . 12 July 2012 . 19 . 3 . 10.37236/2481. free .