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The ternary commutator or ternutator, defined as the alternating sum of the product of three operators, has recently drawn much attention as an interesting structure generalising the commutator. The ternutator satisfies cubic identities…
In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms \begin{align} I_{i,j} = \int_{\mathbb{R}^2}\phi(x)\frac{x_ix_j}{|x|^{2+\alpha}} \d x, \quad…
For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by…
Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that…
An integer of the form $P_8(x)=3x^2-2x$ for some integer $x$ is called a generalized octagonal number. A quaternary sum $\Phi_{a,b,c,d}(x,y,z,t)=aP_8(x)+bP_8(y)+cP_8(z)+dP_8(t)$ of generalized octagonal numbers is called {\it universal} if…
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively.…
Generalized Pascal matrix whose elements are generalized binomial coefficients is included in the group of generalized Riordan arrays. There is a special set of generalized Riordan arrays defined by parameter $q$. If $q=0$, they are…
We consider a generalization of the quaternion ring $\Big(\frac{a,b}{R}\Big)$ over a commutative unital ring $R$ that includes the case when $a$ and $b$ are not units of $R$. In this paper, we focus on the case $R=\mathbb{Z}/n\mathbb{Z}$…
In this paper, we present a new generalization of the Lucas numbers by matrix representation using Genaralized Lucas Polynomials. We give some properties of this new generalization and some relations between the generalized order-k Lucas…
In this paper will be introduced large, probably complete family of complex base systems, which are 'proper' - for each point of the space there is a representation which is unique for all but some zero measure set. The condition defining…
We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using…
As is well-known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate version of such functions and polynomials, degenerate polylogarithm functions were introduced and degenertae…
Two differential calculi are developped on an algebra generalizing the usual q-oscillator algebra and involving three generators and three parameters. They are shown to be invariant under the same quantum group that is extended to a…
The article studies a class of generalized factorial functions and symbolic product sequences through Jacobi type continued fractions (J-fractions) that formally enumerate the divergent ordinary generating functions of these sequences. The…
An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to…
This paper derives sparse recurrence relations between orthogonal polynomials on a triangle and their partial derivatives, which are analogous to recurrence relations for Jacobi polynomials. We derive these recurrences in a systematic…
We report the three main ingredients to calculate three- and four-electron integrals over Gaussian basis functions involving Gaussian geminal operators: fundamental integrals, upper bounds, and recurrence relations. In particular, we…
In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal…
In this paper, we present a new hypercomplex number system, Trinition, that has an unusual structure of commutativity, noncommutativity, nonassociativity, and deformability.
We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists…