Details

Autor(en) / Beteiligte

• Gasper, George
• Rahman, Mizan

Titel

Positivity of the Poisson Kernel for the Continuous q -Ultraspherical Polynomials

Ist Teil von

• SIAM journal on mathematical analysis, 1983-03, Vol.14 (2), p.409-420

Ort / Verlag

Philadelphia: Society for Industrial and Applied Mathematics

Erscheinungsjahr

1983

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• Rogers' [Proc. London Math. Soc., 24 (1893), pp. 117-179] bilinear generating function for the continuous $q$-Hermite polynomials \[ \frac{{\left( {t^2 ;q} \right)_\infty }} {{\left| {\left( {te^{i(\theta + \psi )} ;q} \right)_\infty \left( {te^{i(\theta - \psi )} ;q} \right)_\infty } \right|^2 }} = \sum\limits_{n = 0}^\infty {\frac{{H_n ( {\cos \theta |q} )H_n ( {\cos \psi |q} )}} {{(q;q)_n }}} t^n , \] where $(q;q)_n = (1 - q)(1 - q^2 ) \cdots (1 - q^n )$ and $(a;q)_\infty = (1 - a)(1 - aq)(1 - aq^2 ) \cdots $, is extended to the continuous $q$-ultraspherical polynomials $C_n (x;\beta |q)$ and used to give conditions for the positivity of the Poisson kernel for these polynomials. Related bilinear generating functions are also considered.