

Seminar on Complex Analysis (Gonchar Seminar)
October 29, 2018 17:00–19:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)






About the properties of the zeros of the “perturbed ” exponential function $\sum_{n\geq 0} q^{n(n1)/2}z^n/n!$
A. V. Dyachenko^{} ^{} Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow

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Abstract:
For $q\leq1$ the “perturbed” (also “deformed”/“parametric”) exponential function is a unique analytic solution of the functional equation $f'(z)=f(qz)$ satisfying the initial condition $f(0)=1$. In literature, this
function appeared many times due to its connection to the exponential function. G. Valiron wrote that it is a simplest entire function after $e^x$. It turns to be related to the Tutte polynomials, its power series evidently resembles the power series of the Jacobi thetafunction. Nevertheless, although all zeros of the thetafunction are well known (and simple), we still cannot fully describe the properties that have the zeros of the perturbed exponential function in general case.
In 2009, A. Sokal stated a conjecture that all zeros of the perturbed exponential function are simple (as well as a number of further conjectures). This fact would be useful for studying the dynamics of the zeros as a function of the complex parameter $q$. In this talk, I would like to present the results that I know on this conjecture, as well as a few related questions.

