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In this paper we introduce reproducing kernel Hilbert spaces of polyanalytic functions of infinite order. First we study in details the counterpart of the Fock space and related results in this framework. In this case the kernel function is given by ezw+zw which can be connected to kernels of polyanalytic Fock spaces of finite order. Segal-Bargmann and Berezin type transforms are also considered in this setting. Then, we study the reproducing kernel Hilbert spaces of complex-valued functions with reproducing kernel 1 / ((1 − zw)(1 − zw)) and 1 / (1 − 2Re zw) . The corresponding backward shift operators are introduced and investigated.


This is the initial draft of an article that was later underwent peer review and was accepted for publication in Integral Equations and Operator Theory, volume 94, in 2022. The final publication may differ and is available at Springer via

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This version: the authors. Final, peer-reviewed version: Springer.

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