Szasz Analytic Functions and Noncompact Kähler Toric Manifolds

We show that the classical Szasz analytic function SN (f)(x) is related to the Bergman kernel for the Bargmann-Fock space. Then we generalize this relation to any noncompact toric Kähler manifold, defining the generalized Szasz analytic function ShN (f)(x). Then we will prove the complete asymptotic expansion of ShN (f)(x) and its scaling limit property. As examples, we will compute the generalized Szasz analytic function for the unit ball with the Bergman metric and complex 1,2 and 3-dimensional Kepler manifolds with incomplete Kähler metrics. In addition, three examples about noncompact complete Kähler toric manifolds are discussed, among which are the total space O(−1) → CP, toric Sasaki-manifolds and Reinhardt Domains.