Rogers–Szegő polynomials - Alchetron, the free social encyclopedia

In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Szegő (1926), who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers. They are given by

h n ( x ; q ) = ∑ k = 0 n ( q ; q ) n ( q ; q ) k ( q ; q ) n − k x k

where (q;q)_n is the descending q-Pochhammer symbol.

Furthermore, the h n ( x ; q ) satisfy (for n ≥ 1 ) the recurrence relation

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 .

References

Rogers–Szegő polynomials Wikipedia

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