Related papers: Complex numbers in three dimensions
In a joint paper [4] by Otsubo, Terasoma and the first author, we proved that the special value 3F2(a,b,q;a+b,q;1) of the generalized hypergeometric function is a linear combination of log of algebraic numbers if the triplet (a,b,q) of…
A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1P_3(x_1)+a_2P_3(x_2)+\cdots+a_kP_3(x_k)$ where $a_i's$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it…
In our joint papers [FL1-FL2] we revive quaternionic analysis and show deep relations between quaternionic analysis, representation theory and four-dimensional physics. As a guiding principle we use representation theory of various real…
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms…
In this article, we consider the problem of determining formulas for the number of representations of a natural number $n$ by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the…
We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of…
We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix…
In this paper, we present a mixed-type integral-sum representation of the cylinder functions $\mathscr{C}_\mu(z)$, which holds for unrestricted complex values of the order $\mu$ and for any complex value of the variable $z$. Particular…
We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula…
Let $\mathfrak g$ be an infinite-dimensional Lie algebra and $G$ be the algebraic completion of its module. Using a geometric interpretation in terms of sewing two Riemann spheres with a number of marked points, we introduce a…
This paper begins the study of infinite-dimensional modules defined on bicomplex numbers. It generalizes a number of results obtained with finite-dimensional bicomplex modules. The central concept introduced is the one of a bicomplex…
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,…
The formulation of hypermultiplets that has been developed for 5-dimensional matter multiplets is by dimensional reductions translated into the appropriate spinor language for 6 and 4 dimensions. We also treat the theories without actions…
A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains,…
We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space $\mathbb{E}_3$ with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form $H…
A complex potential is a holomorphic function $\Omega:\mathbb{C} \to \mathbb{C}$ whose real and imaginary parts generate a pair of orthogonal foliations, representing the equipotential lines and the streamlines of $\dot{z} =…
Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…
In multicentric calculus one takes a polynomial $p$ with distinct roots as a new variable and represents complex valued functions by $\mathbb C^d$-valued functions, where $d$ is the degree of $p$. An application is e.g. the possibility to…
We discuss the evaluation of certain d dimensional angular integrals which arise in perturbative field theory calculations. We find that the angular integral with n denominators can be computed in terms of a certain special function, the…
In a recent paper, Adamchik [V.S. Adamchik, On the Hurwitz function for rational arguments, Appl. Math. Comp. 187 (2007) 3--12] expressed in a closed form symbolic derivatives of four functions belonging to the class of functions whose…