共查询到20条相似文献,搜索用时 22 毫秒
1.
Felix Albrecht Kevin A Grasse Nelson Wax 《Journal of Mathematical Analysis and Applications》1981,79(1):178-202
An algebraic setting for the Roman-Rota umbral calculus is introduced. It is shown how many of the umbral calculus results follow simply by introducing a comultiplication map and requiring it to be an algebra map. The same approach is used to construct a q-umbral calculus. Our umbral calculus yields some of Andrews recent results on Eulerian families of polynomials as corollaries. The homogeneous Eulerian families are studied. Operator and functional expansions are also included. 相似文献
2.
Viktor Abramov 《Advances in Applied Clifford Algebras》2007,17(4):577-588
We construct a q-analog of exterior calculus with a differential d satisfying d
N
= 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found. 相似文献
3.
We give an elementary calculus proof of the asymptotic formulas for the zeros of the q-sine and cosine functions which have been recently found numerically by Gosper and Suslov. Monotone convergent sequences of the lower and upper bounds for these zeros are constructed as an extension of our method. Improved asymptotics are found by a different method using the Lagrange inversion formula. Asymptotic formulas for the points of inflection of the basic sine and cosine functions are conjectured. Analytic continuation of the q-zeta function is discussed as an application. An interpretation of the zeros is given. 相似文献
4.
5.
A. K. Kwaśniewski 《Advances in Applied Clifford Algebras》2006,16(1):29-39
One presents here the *ψ-Bernoulli-Taylor* formula of a new sort with the rest term of the Cauchy type recently derived by the author in the case
of the so called ψ-difference calculus which constitutes the representative for the purpose case of extended umbral calculus.
The central importance of such a type formulas is beyond any doubt - and recent publications do confirm this historically
established experience.
*see below: a historical remark based on Academician N. Y. Sonin article published in Petersburg in 19-th century. We owe
this information and the article to Professor O. V. Viscov from Moscow.
**see: C. Graves, On the principles which regulate the interchange of symbols in certain symbolic equations, Proc. Royal Irish Academy 6 (1853-1857), 144-152. 相似文献
6.
In this paper, some irreducible graded modules with 1-dimensional homogeneous spaces over the Virasoro-like algebra and its q-analogs are constructed. The unitarizability of these modules, and the conditions under which two of such irreducible graded modules are ismorphic are determined. Some other kinds of irreducible graded modules with 1-dimensional homogeneous spaces over the Virasorolike algebra and its q-analogs are also given. 相似文献
7.
“A Calculus of Sequences” started in 1936 by Ward constitutes the general scheme for extensions of classical operator calculus
of Rota—Mullin considered by many afterwards and after Ward. Because of the notation we shall call the Ward's calculus of
sequences in its afterwards elaborated form—a ψ-calculus.
The ψ-calculus in parts appears to be almost automatic, natural extension of classical operator calculus of Rota—Mullin or
equivalently—of umbral calculus of Roman and Rota.
At the same time this calculus is an example of the algebraization of the analysis—here restricted to the algebra of polynomials.
Many of the results of ψ-calculus may be extended to Markowsky Q-umbral calculus where Q stands for a generalized difference operator, i.e. the one lowering the degree of any polynomial by one.
This is a review article based on the recent first author contributions [1]. As the survey article it is supplemented by the
short indicatory glossaries of notation and terms used by Ward [2], Viskov [7, 8], Markowsky [12], Roman [28–32] on one side
and the Rota-oriented notation on the other side [9–11, 1, 3, 4, 35] (see also [33]). 相似文献
8.
Let Ω be a vector space over a finite field with q elements. Let G denote the general linear group of automorphisms of Ω and let us consider the left regular representation
associated with the natural action of G on the set X of linear subspaces of Ω. In this paper we study a natural basis B of the algebra EndG(L
2(X)) of intertwining maps on L
2(X). By using a Laplacian operator on Grassmann graphs, we identify the kernels in B as solutions of a basic hypergeometric difference equation. This provides two expressions for these kernels. One in terms
of the q-Hahn polynomials and the other by means of a Rodrigues type formula. Finally, we obtain a useful product formula for the
mappings in B. We give two different proofs. One uses the theory of classical hypergeometric polynomials and the other is supported by
a characterization of spherical functions in finite symmetric spaces. Both proofs require the use of certain associated Radon
transforms. 相似文献
9.
Carnovale 《Constructive Approximation》2008,18(3):309-341
The investigation of a q -analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, with special attention to
the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras
(with respect to the q -convolution), depending on a parameter s>0 , is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and is shown
to be the quotient of an algebra studied in a previous paper modulo the kernel of a q -analogue of the Fourier transform. This result has an analytic interpretation in terms of analytic functions, whose q -moments have a (fast) decreasing behavior and allows the extension of Koornwinder's inversion formula for the q -Fourier transform. A few results on the invertibility of functions with respect to the q -convolution are also obtained and they are applied to the solution of certain simple linear q -difference equations with polynomial coefficients. 相似文献
10.
Noncommutative Galois Extension and Graded <Emphasis Type="Italic">q</Emphasis>-Differential Algebra
Viktor Abramov 《Advances in Applied Clifford Algebras》2016,26(1):1-11
We show that a semi-commutative Galois extension of a unital associative algebra can be endowed with the structure of a graded q-differential algebra. We study the first and higher order noncommutative differential calculus of semi-commutative Galois extension induced by the graded q-differential algebra. As an example we consider the quaternions which can be viewed as the semi-commutative Galois extension of complex numbers. 相似文献
11.
Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions
of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result,
a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek
polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations
of the quantized universal enveloping algebra U
q
(su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson
kernel of Al-Salam and Chihara polynomials.
July 6, 1997. Date accepted: September 23, 1998. 相似文献
12.
Prof. Dr. J. Cigler 《Monatshefte für Mathematik》1981,91(2):105-117
Someq-analogues of the classical Laguerre-polynomials are studied from the point of view of umbral calculus. 相似文献
13.
In 1963 [Ann. of Math. 78, 267-288], Gerstenhaber invented a comp(osition) calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, a graded Lie algebra and a pre-coboundary operator can be associated to every comp system. A derivation deviation of the pre-coboundary operator over the total composition is calculated in two ways, (the long) one of which is essentially new and can be seen as an example and elaboration of the auxiliary variables method proposed by Gerstenhaber in the early days of the comp calculus. 相似文献
14.
Philip Feinsilver 《Acta Appl Math》1990,19(3):207-251
q-Analogs of the basic structures discussed in Lie Algebras and Recurrence Relations I are presented. Theq-Heisenberg-Weyl (qHW) and qsl(2) algebras are discussed in detail. Presently it is known that such structures are very closely tied in with the theory of quantum groups. Among other topics, coherent state representations and their interpretations asq-identities forq-Hermite and Al-Salam-Chihara (q-Meixner) polynomials are discussed. A discussion of Clebsch-Gordan coefficients for a qsu(2)-type algebra is presented. 相似文献
15.
A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach,
we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra
W(p){\mathcal{W}(p)} and for other subalgebras of lattice vertex algebras and their N = 1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural
equivalence between the category of W(p){\mathcal{W}(p)}-modules and the category of modules for the restricted quantum group [`(U)]q(sl2){\overline{\mathcal{U}}_q(sl_2)} , q = e
π
i/p
. We also construct logarithmic representations for a certain affine vertex operator algebra at admissible level realized
in Adamović (J. Pure Appl. Algebra 196:119–134, 2005). In this way we prove the existence of the logarithmic representations
predicted in Gaberdiel (Int. J. Modern Phys. A 18, 4593–4638, 2003). Our approach enlightens related logarithmic intertwining
operators among indecomposable modules, which we also construct in the paper. 相似文献
16.
We prove that a sectorial operator admits an H
∞-functional calculus if and only if it has a functional model of Nagy–Foiaş type. Furthermore, we give a concrete formula
for the characteristic function (in a generalized sense) of such an operator. More generally, this approach applies to any
sectorial operator by passing to a different norm (the McIntosh square function norm). We also show that this quadratic norm
is close to the original one, in the sense that there is only a logarithmic gap between them. 相似文献
17.
In this paper, we define a q-deformation of the Dirac operator as a generalization of the one dimensional q-derivative. This is done in the abstract setting of radial algebra. This leads to a q-Dirac operator in Clifford analysis. The q-integration on
\mathbbRm{\mathbb{R}^m}, for which the q-Dirac operator satisfies Stokes’ formula, is defined. The orthogonal q-Clifford- Hermite polynomials for this integration are briefly studied. 相似文献
18.
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 polynomials of degree n+1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurrence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra. 相似文献
19.
Huang Liping 《Linear and Multilinear Algebra》2013,61(2-3):99-107
Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial qA (λ) of A over F. By using qA (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix. 相似文献
20.
Philip Feinsilver 《Monatshefte für Mathematik》1987,104(2):89-108
Theq-analog and finite difference analog of the canonical Heisenberg-Weyl algebra are studied. The basic representations are found. Analogs of the standard boson Fock space are constructed. Various examples and illustrations are presented. 相似文献