Mathematics – Combinatorics
Scientific paper
2006-12-15
Mathematics
Combinatorics
16 pages, revised version, to appear in J. Phys. A: Math. Theor
Scientific paper
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szeg\"{o} polynomials $h_n(x,y|q)$. The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson kernel formula for the continuous big $q$-Hermite polynomials $H_n(x;a|q)$ due to Askey, Rahman and Suslov. Mehler's formula for $h_n(x,y|q)$ involves a ${}_3\phi_2$ sum and the Rogers formula involves a ${}_2\phi_1$ sum. The proofs of these results are based on parameter augmentation with respect to the $q$-exponential operator and the homogeneous $q$-shift operator in two variables. By extending recent results on the Rogers-Szeg\"{o} polynomials $h_n(x|q)$ due to Hou, Lascoux and Mu, we obtain another Rogers-type formula for $h_n(x,y|q)$. Finally, we give a change of base formula for $H_n(x;a|q)$ which can be used to evaluate some integrals by using the Askey-Wilson integral.
Chen William Y. C.
Saad Husam L.
Sun Lisa H.
No associations
LandOfFree
The Bivariate Rogers-Szegö Polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Bivariate Rogers-Szegö Polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Bivariate Rogers-Szegö Polynomials will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-133241