Related papers: Resurgent Lambert series with characters

The summatory function of a $q$-regular sequence in the sense of Allouche and Shallit is analysed asymptotically. The result is a sum of periodic fluctuations for eigenvalues of absolute value larger than the joint spectral radius of the…

Building on the results of [1,2], we study the resurgence of $q$-Pochhammer symbols and determine their summability and quantum modularity properties. We construct a new, infinite family of pairs of modular resurgent series from the…

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a…

In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic…

We provide estimates for the convolution product of an arbitrary number of "resurgent functions", that is holomorphic germs at the origin of $C$ that admit analytic continuation outside a closed discrete subset of $C$ which is stable under…

Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations…

We introduce a family of L-series specialising to both L-series associated to certain Dirichlet characters over F_q[T] and to integral values of Carlitz-Goss zeta function associated to F_q[T]. We prove, with the use of the theory of…

The temperature inversion properties of the internal energy, E, on odd spheres, and its derivatives, together with their expression in elliptic terms, as expounded in previous papers, are extended to the integrals of E, thence making…

We consider meromorphic transforms given by meromorphic kernels and study their asymptotic expansions under a certain rescaling. Under decay assumptions we establish the full asymptotic expansion in the rescaling parameter of these…

Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are…

This text is about the mathematical use of certain divergent power series. The first part is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the…

We investigate Dirichlet-type series generated by representation functions that count the number of ways an integer can be expressed as a sum of 'k' signed higher even powers. By combining generalized theta generating functions with a…

This article deals with three types of mutually inverse series relating Ferrers and associated Legendre functions of arbitrary complex indexes and orders established on the base of integral representations by using a number of generating…

Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin transform of the symmetric square $L$-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the non-trivial zeros of the Riemann…

The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series.…

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of…

We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in…

We define a compact version of the Hilbert transform, which we then use to write explicit expressions for the partial sums and remainders of arbitrary Fourier series. The expression for the partial sums reproduces the known result in terms…

Building upon the work of Stucker, Vennos, and Young we derive generalized Dedekind sums arising from period integrals applied to holomorphic Eisenstein series attached to pairs of primitive non-trivial Dirichlet characters. Furthermore, we…

The Lagrange inversion formula for power series is one of the classical formulas from analysis and combinatorics. A nice geometric interpretation of this formula in terms of the Stasheff polytopes was discovered by Loday. We show that it…