Reproducing kernels of weighted poly-Bergman spaces on the upper half-plane. (English summary) Bol. Soc. Mat. Mexicana (3) 13 (2007), no. 2, 345–356. In the paper under consideration weighted Bergman spaces of polyanalytic functions on the upper half-plane Π are studied. Let n ≥ 1 be an integer. Recall that a function f is called polyanalytic of order n in an open set U ⊂ C (or, briefly, n-analytic in U) if ∂ n f/∂z n = 0 in U. It follows easily that every n-analytic in U function f has the form f(z) = z n−1 fn−1(z) + · · · + zf1(z) + f0(z), where f0,..., fn−1 are holomorphic functions in U. Now let λ ∈ (−1, ∞). By definition, the weighted polyanalytic Bergman space A2 n,λ (Π) consists of all n-analytic functions belonging to the space L2 (Π, µλ), where µλ denotes the measure (λ + 1)(2y) λ dxdy on Π. The author proves that the orthogonal projection Bn,λ from L2 (Π, µλ) onto A2 n,λ (Π) (this projection is called poly-Bergman projection) admits the integral representation Bn,λf(z) = f(ζ) Kn,λ(z, ζ) dµλ(ζ). Here the reproducing kernel Kn,λ(z, ζ) has the form 1 Kn,λ(z, ζ) = ((z − z) − (ζ − ζ))(iζ − iz) λ+2 where αjkn
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