Multiple sums and integrals associated with planar rooted trees

13.07.2022, 11:00  –  Online Seminar
Arbeitsgruppenseminar Analysis

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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