Related papers: The Bicomplex-Real Calculus and Applications to Bi…
The algebra B of bicomplex numbers is viewed as a complexification of the Archimedean f-algebra of hyperbolic numbers D. This lattice-theoretic approach allows us to establish new properties of the so-called D-norms. In particular, we show…
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-$D$ Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the…
Hypercomplex numbers are unital algebras over the real numbers. We offer a short demonstration of the practical value of hypercomplex analytic functions in the field of partial differential equations.
The deformation bicomplex of a module-algebra over a bialgebra is constructed. It is then applied to study algebraic deformations in which both the module structure and the algebra structure are deformed. The cases of module-coalgebras,…
We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.
Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory,…
We consider and discuss some basic properties of the bicomplex analog of the classical Bargmann space. The explicit expression of the integral operator connecting the complex and bicomplex Bargmann spaces is also given. The corresponding…
In this paper, we are concerned with the bicomplex analog of the well-known result asserting that real-valued harmonic functions, on simply connected domains, are the real parts of holomorphic functions. We show that this assertion, word…
In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a…
We apply the theory of weighted bicategorical colimits to study the problem of existence and computation of such colimits of birepresentations of finitary bicategories. The main application of our results is the complete classification of…
Motivated by derivation of the Dirac type delta-function for quantum states in Fock-Bargmann representation, we find q-binomial expansion in terms of q-Hermite polynomials, analytic in two complex arguments. Based on this representation, we…
This paper delves into classical multiple orthogonal polynomials with an arbitrary number of weights, including Jacobi-Pi\~neiro, Laguerre of both first and second kinds, as well as multiple orthogonal Hermite polynomials. Novel explicit…
The Hermite interpolation formulas are based on the interpretation of interpolation nodes as roots of suitable polynomials. Therefore, such formulas belong to the class of algebraic interpolations. The article considers a multidimensional…
We use generating functions to express orthogonality relations in the form of $q$-beta integrals. The integrand of such a $q$-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal
It is known that Rodrigues formulas provide a very powerful tool to compute orthogonal polynomials with respect to classical weights. We provide an example of bivariate multiple polynomials on the simplex defined via a Rodrigues formula.…
We show some applications of the formulas-as-polynomials correspondence: 1) a method for (dis)proving formula isomorphism and equivalence based on showing (in)equality; 2) a constructive analogue of the arithmetical hierarchy, based on the…
The classical Hermite-Biehler theorem describes possible zero sets of complex linear combinations of two real polynomials whose zeros strictly interlace. We provide the full characterization of zero sets for the case when this interlacing…
In this paper, we investigate the properties of q-Hermite polynomials related to q-Bernstein polynomials. From these properties, we derive some interesting relations between q-Berstein polynomials and q-Hermite polynomials.
We generalize the classical lifting and recombination scheme for rational and absolute factorization of bivariate polynomials to the case of a critical fiber. We explore different strategies for recombinations of the analytic factors,…
We give two widest Mehler's formulas for the univariate complex Hermite polynomials $H_{m,n}^\nu$, by performing double summations involving the products $u^m H_{m,n}^\nu (z,\overline{z}) \overline{H_{m,n}^\nu (w,\overline{w})}$ and $u^m…