Yannic Vargas (Weierstrass Institute)
We investigate and compare two different ways of associating multiple sums and multiple integrals to a certain class of planar rooted trees (Schröder trees). These give rise to characters on the algebra of planar Schröder trees equipped with a shuffle product. They generalise the multiple integrals associated with binary trees considered by Patras and Ebrahimi-Fard and specialise to characters on the Loday-Ronco algebra of binary trees.
A first construction uses the universal property of rooted trees in the spirit of previous work of Clavier, Guo, Paycha and Zhang in the divergent case and Clavier's study in the convergent setup. An alternative approach uses the inclusion-exclusion principle and the multiplicative property of sums and integrals over independent sets of variables. Specifying the class of summands of multiple sums associated to trees leads to branched zeta functions, which generalise multiple zeta functions. We show that they coincide with the planar version of arborified (or branched) zeta functions studied by Clavier in the convergent case. Criteria for convergence follow from a "flatening procedure" using a well-known connection between the algebras of set compositions and of Schröder trees.
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