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We introduce $p$-adic Kummer spaces of continuous functions on $\mathbb{Z}_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and certain decompositions.…
Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the…
The generalized Fermi-Dirac functions and their derivatives are important in evaluating the thermodynamic quantities of partially degenerate electrons in hot dense stellar plasmas. New recursion relations of the generalized Fermi-Dirac…
We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a…
We calculate the least upper bounds of pointwise and uniform approximations for classes of $2\pi$-periodic functions expressible as convolutions of an arbitrary square summable kernel with functions, which belong to the unit ball of the…
The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…
Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Polya frequency functions, or totally positive kernels are treated from a unifying perspective. Besides the stark rigidity of the polynomial…
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different…
We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlev\'e equations and in the study of exceptional orthogonal polynomials. We determine their…
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial…
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…
A mathematical method for constructing fractal curves and surfaces, termed the $p\lambda n$ fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal…
An algebraic version of Kashiwara and Schapira's calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King's numerical conditions for a stratified set of…
Fekete polynomials associate with each prime number $p$ a polynomial with coefficients $-1$ or $1$ except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo $p$. These polynomials were…
We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…
New formulas are given for the grand partition function of paraboson systems of order p with n orbitals and parafermion systems of order p with m orbitals. These formulas allow the computation of statistical and thermodynamic functions for…
Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce…
We obtain a combinatorial formula for the positive integer moments of the partition function of the $C\beta E_{N}$ field, or equivalently the moments of the moments of the characteristic polynomial of the $C\beta E_{N}$ ensemble. We then…
We apply a symbolic approach of the general quadratic decomposition of polynomial sequences - presented in a previous article referenced herein - to polynomial sequences fulfilling specific orthogonal conditions towards two given…
This article is a survey on the topic of polynomial amoebas. We review results of papers written on the topic with an emphasis on its computational aspects. Polynomial amoebas have numerous applications in various domains of mathematics and…