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1.
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+1n+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   

2.
3.
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   

4.
The aim of this paper is to deduce integral and discrete representations of some Hq -semiclassical forms of class one. These forms were studied by B. Bouras and F. Marcellan in [1], (Quadratic decomposition of a family of Hq-semiclassical orthogonal polynomial sequences, J. Di?erence Equ. Appl. 18 (2012), 2039–2057), but unfortunately, the above problem of representation remained opened in that work. In this paper, we find the moments investigate the of theses forms to solve this problem.  相似文献   

5.
In this paper we present a new characterization for the classical discrete and qq-classical (discrete) polynomials (in the Hahn's sense).  相似文献   

6.
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   

7.
In this paper we establish the spectral decomposition of the Berezin transforms on the space Mat(2,) of complex 2×2 matrices related to the two-sided action ofU(2)×U(2). The eigenspaces are described explicitly by means of the matrix elements of a certain representation ofGL(2,) and each eigenvalue is expressed as a finite sum involving the MeijerG-functions evaluated at 1 and the Hahn polynomials.  相似文献   

8.
9.
《Quaestiones Mathematicae》2013,36(8):1083-1116
Abstract

As far as we know, usual computer algebra packages can not compute denumerants for almost medium (about a hundred digits) or almost medium-large (about a thousand digits) input data in a reasonably time cost on an ordinary computer. Implemented algorithms can manage numerical n-semigroups for small input data.

Basically, the denumerant of a non-negative element s ∈ ? is the number of non-negative integer solutions of certain linear non-negative Diophantine equation which constant term is equal to s.

Here we are interested in denumerants of numerical 3-semigroups which have almost medium input data. A new algorithm for computing denumerants is given for this task. It can manage almost medium input data in the worst case and medium-large or even large input data in some cases.  相似文献   

10.
Summary Given an integern 1, and the orthogonal polynomials n (·; d) of degreen relative to some positive measured, the polynomial system induced by n is the system of orthogonal polynomials corresponding to the modified measure . Our interest here is in the problem of determining the coefficients in the three-term recurrence relation for the polynomials from the recursion coefficients of the orthogonal polynomials belonging to the measured. A stable computational algorithm is proposed, which uses a sequence ofQR steps with shifts. For all four Chebyshev measuresd, the desired coefficients can be obtained analytically in closed form. For Chebyshev measures of the first two kinds this was shown by Al-Salam, Allaway and Askey, who used sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here, analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing properties involving the zeros of n and those of are studied for Gegenbauer measures, as well as the orthogonality—or lack thereof—of the polynomial sequence .Work supported in part by the National Science Foundation under grant DMS-9023403.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.  相似文献   

11.
D. E. Loeb and G.-C. Rota, using the operator of differentiation D, constructed the logarithmic algebra that is the generalization of the algebra of formal Laurent series. They also introduced Appell graded logarithmic sequences and binomial (basic) graded logarithmic sequences as sequences of elements of the logarithmic algebra and extended the main results of the classical umbral calculus on such sequences. We construct an algebra by an operator d that is defined by the formula (1.1). This algebra is an analog of the logarithmic algebra. Then we define sequences analogous to Boas-Buck polynomial sequences and extend the main results of the nonclassical umbral calculus on such sequences. The basic logarithmic algebra constructed by the operator of q-differentiation is considered. The analog of the q-Stirling formula is obtained.  相似文献   

12.
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy (i) and (ii) below:
(i)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
(ii)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by
  相似文献   

13.
Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A:VV and A:VV that satisfy (i) and (ii) below:
(i)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
(ii)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
We call such a pair a Leonard pair on V. Let (resp. ) denote a basis for V referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation S:VV that sends v0 to a scalar multiple of vd, fixes w0, and sends wi to a scalar multiple of wi for 1?i?d. We call S the switching element. We describe S from many points of view.  相似文献   

14.
Advances in nanotechnology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices that might be instrumental in the micro-computer industry which are predominantly based on electron transport phenomena. There are many experiments and molecular dynamical simulations which show that a wave is generated on the outer cylinder as a result of the oscillation of the inner carbon nanotube and that the frequency of this wave is also in the gigahertz range. As a preliminary to analyze and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here we determine some new analytical expressions for the van der Waals force using the Lennard–Jones potential for general lengths of the inner and outer tubes. These expressions are utilized together with Newton’s second law to determine the motion of an oscillating inner tube, assuming that any frictional effects may be neglected. An idealized and much simplified representation of the Lennard–Jones force is used to determine a simple formula for the oscillation frequency resulting from an initial extrusion of the inner tube. This simple formula is entirely consistent with the existing known behavior of the frequency and predicts a maximum oscillation frequency occurring when the extrusion length is (L 2L 1)/2 where L 1 and L 2 are the respective half-lengths of the inner and outer tubes (L 1 < L 2).  相似文献   

15.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy (i) and (ii) below:
(i)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
(ii)
There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A is diagonal.
We call such a pair a Leonard pair on V. Let diag(θ0θ1, … , θd) denote the diagonal matrix referred to in (ii) above and let denote the diagonal matrix referred to in (i) above. It is known that there exists a basis u0u1, … , ud for V and there exist scalars ?1?2, … , ?d in K such that Aui = θiui + ui+1 (0 ? i ? d − 1), Aud = θdud, , . The sequence ?1?2, … , ?d is called the first split sequence of the Leonard pair. It is known that there exists a basis v0v1, … , vd for V and there exist scalars ?1?2, … , ?d in K such that Avi = θdivi + vi+1 (0 ? i ? d − 1),Avd = θ0vd, , . The sequence ?1?2, … , ?d is called the second split sequence of the Leonard pair. We display some attractive formulae for the first and second split sequence that involve the trace function.  相似文献   

16.
Abstract. We construct determinantal expressions for the zonal spherical functions on the hyperboloids with p,q odd (and larger than 1). This gives rise to explicit evaluation formulas for hypergeometric series representing half-integer parameter families of Jacobi functions and (via specialization) Jacobi polynomials. Received November 18, 1999 / Published online October 30, 2000  相似文献   

17.
It is known that if (A,A*) is a Leonard pair, then the linear transformations A, A* satisfy the Askey-Wilson relations
  相似文献   

18.
Let V denote a vector space with finite positive dimension, and let (AA) denote a Leonard pair on V. As is known, the linear transformations A, A satisfy the Askey-Wilson relations
  相似文献   

19.
《Quaestiones Mathematicae》2013,36(7):985-1003
Abstract

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial di?erential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function 1F2(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function 1F2(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.  相似文献   

20.
 Let K be a field of characteristic 0 and let p, q, G 0 , G 1 , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G n (x)) n=0 be defined by the second order linear recurring sequence
In this paper we give conditions under which the diophantine equation G n (x) = G m (P(x)) has at most exp(1018) many solutions (n, m) ε ℤ2, n, m ⩾ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt [14]. Under the same conditions we present also bounds for the cardinality of the set
In the last part we specialize our results to certain families of orthogonal polynomials. This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT. The second author was supported by the Hungarian National Foundation for Scientific Research Grants No 16741 and 38225. Received June 5, 2001; in revised form February 26, 2002 RID="a" ID="a" Dedicated to Edmund Hlawka on the occasion of his 85th birthday  相似文献   

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