Related papers: A shifted binomial theorem and trigonometric serie…

Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted…

A weighted sums of squares decomposition of positive Borel measurable functions on a bounded Borel subset of the Euclidean space is obtained via duality from the spectral theorem for tuples of commuting self-adjoint operators. The analogous…

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space.…

In this paper we consider a transformation $L_a$ of sequences of complex numbers. We find the inverse transformation of $L_a$ as well as the inverse of a related transformation $\tilde{L}_a$. We explore a connection to the binomial…

We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion…

I reconsider the approximation of Bessel functions with finite sums of trigonometric functions, in the light of recent evaluations of Neumann-Bessel series with trigonometric coefficients. A proper choice of the angle allows for an…

We afford the problem of counting the blocks of a given length made with symbols drawn from an alphabet and relate this number to Fibonacci-like recurrent relations. The recurrence polynomia allows to calculate the limit ratio of two…

We prove the equivalence of the frame property and the closedness for a weighted shift-invariant space. We also construct a sequence $\Phi^{2k+1}$ and the sequence of spaces $V^p_\mu(\Phi^{2k+1})$, $k\in{\mathbb{N}}$, on $\mathbb{R},$ with…

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…

In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as…

We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…

Using general principles in the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an…

We study the shifted convolution sum of the divisor function and some other arithmetic functions.

We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair $(\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))$. It is inspired by a similar approach of Kieburg and K\"osters…

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We…

Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we…

In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and…

Modular operads relevant to string theory can be equipped with an additional structure, coming from the connected sum of surfaces. Motivated by this example, we introduce a notion of connected sum for general modular operads. We show that a…

We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…