Integral Transform and Segal-Bargmann Representation Associated to q-Charlier Polynomials

Abstract

Let ${\mu }_p^{(q)}$ be the q-deformed Poisson measure in the sense of Saitoh Yoshida and ${\nu }_p$ be the measure given by Equation \eqref{eq:nu-q}. In this short paper, we introduce the q-deformed analogue of the Segal-Bargmann transform associated with ${\mu }_p^{(q)}$. We prove that our Segal-Bargmann transform is a unitary map of $L^2({\mu }_p^{(q)})$ onto the q-deformed Hardy space ${\mathcal{H}}^2({\nu }_q)$. Moreover, we give the Segal-Bargmann representation of the multiplication operator by $x$ in $L^2({\mu }_p^{(q)})$, which is a linear combination of the q-creation, q-annihilation, q-number, and scalar operators.

Accepted for the publication in "Quantum Information IV", T. Hida and K. Saito (eds.), World Scientific. Minor misprints have been fixed. Reference information has been updated