The research of the rst author was partially supported by Comision Interministerial de Ciencia y Tecnologa (CICYT) of Spain under grant PB 93-0228-C02-01. The second author is grateful to the Instituto de Fsica, UNAM, Mexico and Consejo Nacional de Ciencia y Tecnologa(CONACyC) for their nancial support. The authors are thankful to Professors J.S. Dehesa, F. Marcellan, A.F. Nikiforov, V.N. Tolstoy and A. Ronveaux for valuable discussions and comments and also to the referees for pointing out some references.
A detailed study of this family was done in  and their main characteristics are given in Table 2 of . Let us now study the duality properties of theq-Racah polynomials. First of all, notice that all the characteristics of these polynomials transform into the corresponding ones by replacing theq-numbers [m] with the standard ones m and theq-Gamma functions e Γ q (x), with the classical ones Γ(x). Therefore, it is reasonable to expect that all the results in Sec. 3 can be extended to this case just replacing the standard numbers and functions by their symmetric q-analogs. We will show only the details for the first case, since the other three are equivalent and we will include only the final result.
In this Section we will use the theorems obtained inthe two previous sections to investigate the spectral properties of several known families of orthogonal q-polynomials. Let us make the observation that for a nite polynomial sequence (e.g. Hahn, Racah and Kravchuk polynomials), i.e., when the degree n of the polynomial is bounded by a xed parameter N (not to be confused with the same letter previously used as generic degree of polynomials), it is assumed that N is suciently large and 1 << n N so that Eq. (61) be fullled.
Inthe last years the study of such polynomials have attracted an in- creasing interest (see e.g. [4, 8, 19, 28] and the references therein) with a special emphasis on the case when the starting functional u is a classical continuous linear functional (this case leads to the Jacobi-Krall, Laguerre- Krall, Hermite-Krall, and Bessel-Krall polynomials, see e.g. [7, 13, 14, 18, 23, 25]) or a classical discrete one (this leads to theHahn-Krall, Meixner- Krall, Kravchuk-Krall, and Charlier-Krall polynomials, see e.g. [6, 7, 15]). Moreover, in  a general theory was developed for modifications of quasi- definite linear functionals that covers all the continuous cases mentioned above whereas in  the case when u is a discrete semiclassical or q-semiclassical linear functional was considered in detail. But in  (see also ) only the linear type lattices (for a discussion on the linear type lattices see ) were considered. Here we go further and study the Krall-type polynomials ob- tained by adding delta Dirac functionals to the discrete functionals u defined on theq-quadratic lattice x(s) = c 1 q s + c 2 q −s + c 3 .
The structure of the paper is as follows. In Section 1 we introduce some notations and de1nitions useful for the next ones. In Section 2, starting from the distributional equation (u) = u that the moment functional u, with respect to which the polynomial sequence is orthogonal, satis1es we will obtain 1ve di<erent characterizations of these q-polynomials. They are quoted in Theo- rems 2:1 and 2:2 and Propositions 2:9 and 2:10, respectively. In Section 3, we deduce the main characteristics of theq-polynomialsin terms of the coeMcients of thepolynomials and of the distributional equation, i.e., the coeMcients of the three-term recurrence relations and of the other characterization relations (those proved in Section 2). In Section 4, all q-classical, according to the Hahn’s de1nition, families of polynomials of theq-Askey Tableu are studied in details including all their characteristics.
An orthogonal polynomial family that generalizes the Racah coefficients or 6j-symbols (so-called Racah and q-Racah polynomials) was introduced in . These polynomials are at the top of the so-called Askey scheme (see, e.g., ) that contains all classical families of hypergeometric orthogonal polynomi- als. Some years later the same authors  introduced the celebrated Askey–Wilson polynomials. The important property of these polynomials is the possibility to obtain from them all known families of hypergeometric polynomials and q-polynomials as particular or limit cases (the review is done inthe nice survey ). The main tool of [6, 7] was the hypergeometric and basic series, respectively. On the other hand, in  (see also ) q-polynomials were considered as the solution of a second-order difference equation of the hypergeometric type on the nonlinear lattice,
It is well known that the classical families of Jacobi, Laguerre, Hermite, and Bessel polynomials are characterized as eigenvectors of a second order linear differential operator with polynomial coefﬁcients, Rodrigues formula, etc. In this paper we present a uniﬁed study of the classical discrete polynomials and q-polynomials of theq-Hahn tableau by using the difference calculus on linear-type lattices. We obtain in a straightforward way several characterization theorems for the classical discrete and q-polynomials of the “q-Hahn tableau”. Finally, a detailed discussion of a characterization by Marcellán et al. is presented.
In order to support the analytical results of the mean- field approach, we have also performed Lanczos ED cal- culations on finite systems with 24 spins and periodic boundary conditions for S = 1/2. The bilayer structure of thelattice makes particularly difficult to study small systems because there are four sites per unit cell. In par- ticular, correlation functions between spins belonging to the same layer can be studied only for a few neighbors. Fig. 4 shows the spin-spin correlation between spins be- longing to the same layer inthe zig-zag direction obtained by SBMFT corresponding to the points (a), (b), (c) and (d) of Fig. 3 for a 10000 sites system. The insets corre- spond to the results obtained for the same points with Lanczos technique on a 24 sites system. Although corre- lations are calculated only for a few sites with Lanczos, the absence of antiferromagnetic order inthe insets of Figures 4.c and 4.d is clear. This is consistent with the SBMFT results corresponding to the main Figures 4.c and 4.d.
This kind of perturbations can be seen as the simplest (lower order) discrete-continuous version of the “bispectral” property. The continuous “bispectral problem” consists of describing all Schr¨odinger type differential operators of second order such that their eigenfunctions should satisfy a dif- ferential equation (of arbitrary finite order) inthe spectral variable. The basic tools are based inthe Darboux factorization method . Inthe discrete- continuous case, this factorization is related to the LU and UL factorization of the Jacobi matrix associated with the sequence of orthogonal polynomials which are the eigenfunctions of a second order linear differential operator of hypergeometric type. Thus the so-called Krall polynomials appear in a natural way [15, 16, 17, 18, 19].
siderably larger than ours. For the unfrustrated case, all the mean field approaches are quite inaccurate compared with much more controlled techniques like QMC. The difference inthe M (Q) values of about 10%, provides, inthe absence of any other quantitative evidence for the accuracy of the method as applied to this model, an indi- cation of the accuracy of the method and of all the results quoted that depend on the order parameters, including the phase boundaries. However, the mean field approach is still very useful to study gapped phases in frustrated systems. On one hand it is well known that for frus- trated systems QMC presents the famous sign problem. On the other hand, the study of quantities like energy gap requires the study of big sizes clusters and the use of exact diagonalization for small size clusters makes it very difficult to extrapolate the results.
In Section 2 we include all the properties of the Hermite and Gegenbauer polynomials which will need. In Section 3 we study the generalized Hermite polynomials and Section 4 is devoted to the Gegenbauer case. In particular, we obtain their expression in terms of the classical polynomials, the hypergeometric representations, the ratio asymptotics, the second order dierential equation and the three-term recurrerence relation that such generalized polynomials satisfy.
Due to the specific application of 780 nm laser in continuous terahertz generation and also differential absorption spectroscopy , another type of external cavity tuning setup named dual mode (dual wavelength) has been developed. In this configuration, first, the initial output of the laser is separated into two beams by adding a beam splitter to the single mode Littorow setup, then each arm is tuned separately based on the rotation of the diffraction gratings. The wavelength of these two tuned modes can be either close or far from each other. The output power of dual mode setup is similar to the single mode one. The schematic setup of dual mode Littow configuration is shown in figure 3.
The classic microstrip radiators, such as patches, radiate with linear polarisation (LP), although circular polarisation (CP) with these elements can also be achieved when two orthogonal modes are excited with the necessary phase difference. Never- theless, arrays of linearly polarised elements are normally preferable, since they allow better polarisation purity, higher gain and radiation efficiency to be obtained. To achieve CP with linearly polarised elements, the sequential rotation technique can be used . In this approach, N radiating elements are sequentially rotated and fed with a phase shift equal to the geometrical rotation angle 360 N o . There are multiple examples of patch arrays with CP that use this technique; however, most of them have a corporate feeding structure, which makes the use of additional circuitry mandatory to provide the necessary current phase shift between elements, thus increasing the com- plexity and cost of the antenna. On the other hand, series feeding with a transmission line has the advantage that the necessary phase shifts between elements are automatic- ally generated if the feeding-line lengths are appropriately designed and more compact structures can be obtained, such in .
Since B(f ) is a graded module, we can consider the graded pieces one at a time. Elements in B(f ) are equivalence classes, so keeping in mind the three expressions above we just take one representative of each class and get B 0 (f ) = C h1i. For degrees greater than one we have that if degree = d = 3s, s ≥ 1, then B d (f ) = hxyz d−2 , z d i; otherwise B d (f ) = hxz d−1 , yz d−1 , z d i, so the monomial basis for B(f)