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# Communications in Number Theory and Physics

## Volume 15 (2021)

### Number 1

### Gamma functions, monodromy and Frobenius constants

Pages: 91 – 147

DOI: https://dx.doi.org/10.4310/CNTP.2021.v15.n1.a3

#### Authors

#### Abstract

In an important paper [8], Golyshev and Zagier introduce what we will refer to as *Frobenius constants* $\kappa_{\rho,n}$ associated to an ordinary linear differential operator $L$ with a reflection type singularity at $t = c$. For every other regular singularity $t = c^\prime$ and a homotopy class of paths $\gamma$ joining $c^\prime$ and $c$, constants $\kappa_{\rho,n} = \kappa_{\rho,n} (\gamma)$ describe the variation around $c$ of the Frobenius solutions $\phi_{\rho,n} (t)$ to $L$ defined near $t = c^\prime$ and continued analytically along $\gamma$. (Here $\rho \in \mathbb{C}$ are local exponents of $L$ at $t = c^\prime$, see Definition 22 below.) Golyshev and Zagier show in certain cases that the $\kappa_{\rho,n}$ are periods, and they raise the question quite generally how to describe the $\kappa_{\rho,n}$ motivically.

The purpose of this work is to develop the theory (first suggested to us by Golyshev) of *motivic Mellin transforms or motivic gamma functions*. Our main result (Theorem 30) relates the generating series $\sum^{\infty}_{n=0} \kappa_{\rho,n}(s-\rho)^n$ to the Taylor expansion at $s = \rho$ of a generalized gamma function, which is a Mellin transform of a solution of the dual differential operator $L^\lor$. It follows from this that the numbers $\kappa_{\rho,n}$ are always periods when $L$ is a geometric differential operator (Corollary 31).

Work of the second author was supported by the National Science Centre of Poland (NCN), grant UMO-2016/21/B/ST1/03084.

Received 28 October 2019

Accepted 9 September 2020

Published 4 January 2021