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We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in…

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the…

We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used…

Inspired by Menshov's representation theorem, we prove that there exists a sequence of frequecies such that any measurable (complex valued) function on R can be represented as a sum of almost everywhere convergent trigonometric series with…

Let $\Phi:V\to V\otimes U$ be an intertwining operator between representations of a simple Lie algebra (quantum group, affine Lie algebra). We define its generalized character to be the following function on the Cartan subalgebra with…

We find linear (as well as quadratic) relations in a very large class of T-functions. The relations may be used in analysis of T-function-based stream ciphers.

We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a…

We give necessary and sufficient condition so that we have d-hypercyclicity for operators who map a holomorphic function to a partial sum of the Taylor expansion. This problem is connected with doubly universal Taylors series and this is an…

We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ''conformal vertex algebra'' or even more generally,…

While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder and Tao have introduced generalized lattice operations based on the projection onto the cone of squares. In two recent papers of the authors…

Generalizations of classical theta functions are proposed that include any even number of analytic parameters for which conditions of quasi-periodicity are fulfilled and that are representations of extended Heisenberg group. Differential…

The aim of this paper is to give the text of a recent introduction to nonlinear generalized functions exposed in my talk in the congress gf2011, which was asked by several participants. Three representative topics were presented: two…

We present a method for constructing global analytical expressions that approximate a function over its entire range. These approximations not only mirror the original function as accurately as desired, but are purposefully created to…

We present an application of the basic mathematical concept of complex functions as topological solitons, a most interesting area of research in physics. Such application of complex theory is virtually unknown outside the community of…

This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…

Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions,…

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying $f(a)-f(b)\equiv0 \pmod {(a-b)}$ for all $a>b$. We characterize this class of functions…

Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Ces\`aro summability of Fourier series, degree of approximation and best approximation by trigonometric…

We investigate subclasses of generalized Bernstein functions related to complete Bernstein and Thorin-Bernstein functions. Representations in terms of incomplete beta and gamma as well as hypergeometric functions are presented. Several…

Based on two point spline approximations of arbitrary order, a series of functions that define lower bounds for sin(x) and sin(x)/x, over the interval [0,Pi/2], with increasingly low relative errors and smaller relative errors than…