Abstract
In this paper, we study the Martin integral representation for nonharmonic functions in discrete settings of infinite homogeneous trees. Recall that the Martin integral representation for trees is analogs to the mean-value property in Euclidean spaces. In the Euclidean case, the mean-value property for nonharmonic functions is provided by the Pizzetti (and co-Pizzetti) series. We extend the co-Pizzetti series to the discrete case. This provides us with an explicit expression for the discrete mean-value property for nonharmonic functions in discrete settings of infinite homogeneous trees.
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Acknowledgments
The authors are grateful to Prof. Woess for constant attention to this work and to the unknown reviewers for providing useful references regarding Pizzetti series and other useful remarks.
Funding
This first author was supported by the Austrian Science Fund (FWF): W1230, Doctoral Program “Discrete Mathematics.” The second author was partially supported by EPSRC grant EP/N014499/1 (LCMH).
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Boiko, T., Karpenkov, O. Martin Integral Representation for Nonharmonic Functions and Discrete Co-Pizzetti Series. Math Notes 106, 659–673 (2019). https://doi.org/10.1134/S0001434619110014
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DOI: https://doi.org/10.1134/S0001434619110014