The structure of the paper is as follows. In Section 2 some necessary results on classical polynomials are collected. In section 3 the factorization of the hypergeometric-type differ- ence equation is discussed, which is used in section 4 to construct a dynamical symmetry algebra in the case of the Charlier polynomials. In section 5 the Kravchuk and the Meixner cases are considered in detail. Finally, in section 6 we briefly discuss a possibility of applying this technique to the -case.
Inspired by the appearance of Macfarlane’s  and Biedenharn’s  important construc- tions of q-analogues of quantum harmonic oscillator, this technique of factorization of difference equations was later employed in a number of publications – in order to study group theoretic properties of the various well-known families of orthogonal polynomials, which can be viewed as q-extensions of the classical Hermite polynomials. So our purpose here is to formu- late a unified approach to deriving all of these results, which correspond to the q-linear spectrum. An important aspect to observe at this point is that we shall mainly (except for the exam- ples in subsection 4.2) confine our attention to those families of q-polynomials, which satisfy discrete orthogonality relation of the type (5). The explanation of such preference is that the factorization of difference equations for instances of q-polynomials with continuous orthogonality property has been already thoroughly studied in –. Observe also that our approach still remains valid in the limit as q → 1; so classical counterparts of q-polynomials, which will be discussed in this paper, are in fact incorporated as appropriate limit cases. But the reader who desires to know more about the factorization in the cases of classical orthogonal polynomials (such as the Kravchuk, Charlier, Meixner, Meixner–Pollaczek, and Hahn) may be referred to [17, 18] and references therein.
Concluding remarks. In this paper we present a simple, unified and con- structive approach for finding linear recurrence relations for the difference hypergeometric-type functions, i.e., solutions of the hypergeometric differ- ence equation (2.1), and apply the general results to some discrete mod- els (e.g. discrete oscillators). Furthermore, the method described here is valuable for new situations, such as higher order recurrence relations and ladder-type relations for the classical discrete orthogonal polynomials.
Let us point out here that the theory of orthogonal polynomials on the non-uniform lattices is based not on the Pearson equation and on the hypergeometric-type di<erence equation of the non-uniform lattices as it is shown in papers [7,26,28] and obviously it is possible to derive many properties of the q-classical polynomials from this di<erence hypergeometric equation. Our purpose is not to show howfrom the di<erence equation many properties can be obtained, but to showthat some of them characterize the q-classical polynomials, i.e., the main aim is the proof of several characterizations of these q-families as well as the explicit computations of the corresponding coeMcients in a uni1ed way. Some of these results on characterizations (e.g. the Al-Salam-Chihara or Marcell+an et al. characterization for classical polynomials) are completely new as far as we know.
Abstract. We show how, using the constructive approach for special functions introduced by Nikiforov and Uvarov, one can obtain recurrence relations for the hypergeometric-type functions not only for the continuous case but also for the dis- crete and q-linear cases, respectively. Some applications in Quantum Physics are discussed.
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) in the spectral variable. The basic tools are based in the Darboux factorization method . In the 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 hypergeometrictype. Thus the so-called Krall polynomials appear in a natural way [15, 16, 17, 18, 19].
computations of these functions. Furthermore, the method presented here is also valid for other quantum systems such as, for instance, the Morse problem  and the relativistic hydrogen atom , since the corresponding wavefunctions are proportional to the Laguerre polynomials. Obviously, this method for finding recurrence relations can be extended to any quantum system whose (radial) wavefunction is proportional to hypergeometric-type functions (see, e.g., ).
eﬀective-mass and envelope-function Hamiltonian which allows us to model sophisticated geometries. The contributions coming from the dielectric mismatch are accounted for using a numerical procedure, and the electron-hole correlations –which are important for long NRs– are treated by carrying full conﬁguration interaction (FCI) calculations. Our results show that in semiconductor NRs the dielectric conﬁnement modiﬁes the energy and intensity of the exciton photoluminescence. The inﬂuence is particularly important in type-II NRs, where the asymmetry between the electron and hole charge distribution enables strong dielectric mismatch eﬀects. In this kind of structures, the electronic density shows a striking response to changes in the dielectric constant of the environment. In insulating environments, the enhanced electron-hole attraction moves the electron density from the center of the NR to the CdTe/CdSe interfaces. Last, we study the eﬀect of longitudinal electric ﬁelds on the excitonic states of the NRs. Our results show that a threshold ﬁeld is required to separate electrons from holes. The value of this critical ﬁeld is strongly dependent on the dielectric constant of the environment.
In this section we have enclosed some formulas for the classical Jacobi polynomials which will be useful to obtain some properties of the Sobolev-type orthogonal polynomials. All the formulas as well as some special properties for the classical Jacobi polynomials can be found in the literature [23, Chapter 1-2], . In this work we will use monic polynomials, i.e., polynomials with leading coecient equal to 1.
In the cluster NGC 6218 (M 12), [Fe/H] = − 1.31 ± 0.028 dex, Carretta et al. (2007) found variations in the sodium abundance near the red giant branch (RGB) bump, which the authors believe to be a reflection of different initial helium content in the target stars. The excess of sodium may indicate that a fraction of BL Her variables are helium overabundant. The masses of the BL Her variables are about 0.50–0.60 M (Bono, Caputo & Santolamazza 1997, with a progenitor mass value of the order of 0.8 M ). In Fig. 9, we present the evolutionary tracks by Chantereau et al. (2016) for the stars with an initial mass of 0.8 M and metal- licity [Fe/H] = − 1.75 calculated for different helium abundances (Y = 0.248, 0.30, 0.37, 0.425, 0.45). The figure shows the posi- tion of the instability strip, as well as the region occupied by the RR Lyrae variables. As can be seen from the figure, stars with some spread in helium abundance (Y = 0.25–0.35) can fall into the region of the instability strip, where the BL Her variables are located. This fact may be responsible for the rather small number of the known variables of this type.
The present work is motivated by the use of friezes to compute cluster variables and is inspired by the result in  giving an explicit formula as a product of 2 × 2 matrices for all cluster variables in coefficient-free cluster algebras of type A , thus explaining at the same time the Laurent phenomenon and positivity.
acceptance rates of 90.73% and 69.70%, respectively. However, acceptance of the PIs communicated by both methods, verbally and in writing, was 76.54%. The differences between degree of acceptance and the type of communication were found to be statistically significant (P<.005) (Table 4). Verbal recommendations were made directly to doctors in 49% of the cases (acceptance rate of 89.19%), to nursing staff (acceptance rate of 92%) and 12% to both doctors and nursing staff (94% acceptance rate). Written communications were issued using the hospital’s computer system (54.55%), by writing in the patient’s medical chart (15.15%), or using both methods (30.30%). Of the recommendations made using verbal and written channels simultaneously, 12.35% were directed to doctors (acceptance rate of 90%), 80.25% to nursing staff (acceptance rate of 73.85%) and 7.41% to both groups (acceptance rate of 83.33%).
As I tried to explain in a previous work (Martín-Jiménez, 2004), and will try to show in more detail in a forthcoming book (Martín- Jiménez, 2015), the establishment of the textual model proposed allows us to refute the idea adequately discussed in contemporary literary studies, that literature is fiction and that without fiction there would not have been literature. In this respect, I consider it essential to distinguish between the act of pretending and of creating fiction. The term pretending is related to European languages (Fr. feindre; Sp. fingir; Port. fingir; It. fingere…) as much as the act of pretending or lying in real life as a creation of fictional works, but both are radically different requiring a precise and clear definition. It is one thing for an author to pretend to feel something that he doesn’t feel and another to create fiction. For this reason, we could consider that an author pretends when exposing something false or made up without developing the characters’ world, while creating fiction when showing a world of the characters governed by the type II and III models of the world.
Polynomials orthogonal with respect to measures which are more general than weight func- tions appear as eigenfunctions of a fourth order linear dierential operator with polynomial coecients. This spectral approach leads to Laguerre-type, Legendre-type and Jacobi-type polynomials introduced by H.L.Krall .
Abstract: Polynuclear transition metal nitrido complexes constitute a class of molecular cage compounds with fascinating structures and interesting bonding properties. However, there is a lack of systematic strategies for the rational construction of aggregates with desired structure and composition. This article provides a brief overview of the structure and bonding modes of polynuclear nitrido complexes, the most common synthetic approaches used to generate such aggregates, and a systematic review of the development of a family of heterometallic nitrido complexes with [MTi 3 N 4 ] cube-type cores. The
We consider the modications of the monic Hermite and Gegenbauer polynomials via the addition of one point mass at the origin. Some properties of the resulting polynomials are studied: three-term recurrence relation, dierential equation, ratio asymptotics, hypergeometric representation as well as, for large n , the behaviour of their zeros.