Diagonals of Laurent series of rational functions and their integral representations

Generating functions naturally split into nested classes: rational, algebraic, and D-finite [1]. Consider a rational function of n complex variables and its Laurent series expansion centered at the origin. The generating function of the subsequence obtained by restricting the sequence of coefficients of this Laurent series to a certain sublattice is called the diagonal of the Laurent series. This construction yields a rich family of functions widely represented in enumerative combinatorics [2], mathematical physics [3], and statistical physics [4]. In our talk, we discuss how integral representations for diagonals help determine their place within the mentioned hierarchy and describe their singular points and branching behavior.
The study was supported by the Russian Science Foundation, project no. 24-21-00217