The structure ofthe paper is as follows. In Section 2, we provide the basic propertiesoftheclassicalorthogonalpolynomialsofdiscrete variable which will be needed, as well as the main data forthe Meixner, Kravchuk and Charlier polynomials. In Section 3 we deduce expressions ofthe generalized Meixner, Kravchuk and Charlier polynomials and its rst dierence derivatives, as well as their representation as hypergeometric functions in the direction raised by Askey. In Section 4, we nd the second order dierence equation which these generalized polynomials satisfy. In Section 5, from the three term recurrence relation (TTRR) oftheclassicalorthogonalpolynomials we nd the TTRR which satisfy the perturbed ones. In Section 6, from the relation ofthe perturbed polynomials P An (x) as a linear combination oftheclassical ones, we nd the tridiagonal matrices associated with the perturbed monic orthogonal polinomial sequence (PMOPS) f P An (x) g
The structure ofthe paper is as follows. In Section 2 we list some ofthe main propertiesoftheclassical Bessel polynomials which will be used later on. In Section 3 we dene the generalized polynomials and nd some of their properties. In Section 4 we obtain the representation ofthe generalized Bessel polynomials in terms ofthe hypergeometric functions. In Section 5 we obtain an asymptotic formula for these polynomials and in Section 6 we establish their quasi-orthogonality. Finally, in Sections 7 and 8 we obtain the three-term recurrence relation that such polynomials satisfy as well as the corresponding Jacobi matrices.
It can be shown (see e.g., [3,8,25]) that the only families satisfying the above deﬁnition are the Hermite, Laguerre, Jacobi, and Bessel polynomials. Nevertheless there are other properties characterizing such families and that can be used to deﬁne theclassical OPS. The oldest one is the so called Hahn characterization—unless this was ﬁrstly observed and proved forthe Jacobi, Laguerre, and Hermite polynomials by Sonin in 1887. In , Hahn proved the following, Theorem 1.2 (Sonin–Hahn [12,19]). A given sequence oforthogonalpolynomials (P n ) n , is a classical sequence if
The term moment problem was used forthe first time in T. J. Stieltjes’ clas- sic memoir  (published posthumously between 1894 and 1895) dedicated to the study of continued fractions. The moment problem is a question in classical analysis that has produced a rich theory in applied and pure mathematics. This problem is beautifully connected to the theory oforthogonalpolynomials, spectral representation of operators, matrix factorization problems, probability, statistics, prediction of stochastic processes, polynomial optimization, inverse problems in financial mathematics and function theory, among many other areas. In the ma- trix case, M. Krein was the first to consider this problem in , and later on some density questions related to the matrix moment problem were addressed in [14, 15, 24, 25]. Recently, the theory ofthe matrix moment problem is used in  forthe analysis of random matrix-valued measures. Since the matrix moment problem is closely related to the theory of matrix orthogonalpolynomials, M. Krein was the first to consider these polynomials in . Later, several researchers have made contributions to this theory until today. In the last 30 years, several known propertiesoforthogonalpolynomials in the scalar case have been extended to the matrix case, such as algebraic aspects related to their zeros, recurrence relations, Favard type theorems, and Christoffel–Darboux formulas, among many others.
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 propertiesofthe q-Racah polynomials. First of all, notice that all the characteristics of these polynomials transform into the corresponding ones by replacing the q-numbers [m] with the standard ones m and the q-Gamma functions e Γ q (x), with theclassical 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 forthe first case, since the other three are equivalent and we will include only the final result.
Theclassicalorthogonalpolynomials are very interesting mathematical objects that have attracted the attention not only of mathematicians since their appearance at the end ofthe XVIII century connected with some physical problems. They are used in several branches of mathematical and physical sciences and they have a lot of useful properties: they satisfy a three-term recurrence relation (TTRR), they are the solution of a second order linear differential (or difference) equation, their derivatives (or ﬁnite differences) also constitute an orthogonal family, their generating functions can be given explicitly, among others (for a recent review see e.g. ). Among such properties, a fundamental role is played by the so-called characterization theorems, i.e., such properties that completely deﬁne and characterize theclassicalpolynomials. Obviously not every property characterize theclassicalpolynomials and as an example we can use the TTRR. It is well-known that, under certain conditions—by the so-called Favard Theorem (for a review see )—, the TTRR characterizes theorthogonalpolynomials (OP) but there exist families of OP that satisfy a TTRR but not a linear differential equation with polynomial coefﬁcients, or a Rodrigues-type formula, etc. In this paper we will complete the works [3,10] proving a new characterization fortheclassicaldiscrete [3,6] and the q-classical [4,10] polynomials. Forthe continuous case see [8,9].
as a modication ofthe rst ones troughtout the addition of two mass points. All the formulas fortheclassical Hahn polynomials can be found in a lot of books ( see for instance the excellent monograph OrthogonalPolynomials in DiscreteVariables by A.F. Nikiforov, S. K. Suslov, V. B. Uvarov , Chapter 2.)
In this Section we will study limit relations involving the modications ofthe Jacobi and Laguerre polynomials as well as the modications oftheclassicalpolynomialsofdiscretevariables. In some way we will obtain an analogue ofthe Askey-scheme of hypergeometric polynomials (for a review see ). Results are predictible but we have found nothing of this kind in the literature. Anyway, we want to remark that the main dierence with respect to theclassical case is the fact, as we will show below, that the point masses change.
In this work we will present a di<erent approach: It can be considered a pure algebraic approach and constitutes an alternative to the two previous ones, and, in some sense is the continuation ofthe Hahn’s work . Furthermore, we will prove here that the q-classicalpolynomials are characterized by several relations, analogue to the ones satis1ed by theclassical “continuous” (Jacobi, Bessel, Laguerre, Hermite) and “discrete” (Hahn, Meixner, Kravchuk and Charlier) orthogonalpolynomials [1,13,21,22] and references therein. Besides, our point of viewis very di<erent from the previous ones based onthe basic hypergeometric series and the di<erence equation, respectively. In fact we start with the distributional equation that the q-moment functionals satisfy and we will prove all the other characterizations using basically the algebraic theory developed by Maroni . So, somehow, this paper is the natural continuation ofthe study started in [22,13] forthe “continuous” and “discrete” orthogonalpolynomials, respectively. Another advantage of this approach is the uni1ed and simple treatment ofthe q-polynomials where all the information is obtained from the coeMcients ofthepolynomials and ofthe distributional or Pearson equation (compare it with the method by the American school  or the Russian ones ).
In this paper we will consider two algorithms which allow us to obtain second order linear dierence equations for certain families ofpolynomials. The corresponding algorithms can be implemented in any computer algebra system in order to obtain explicit expressions ofthe coecients ofthe dierence equations.
In this paper, we have adapted some tools ofthe dynamical analysis of multivariate real discrete problems to analyze the stability ofthe fixed points of iterative methods with memory on quadratic polynomials. As far as we know, this kind of analysis has not been performed before on iterative methods with memory for solving nonlinear equations. From the well-known Secant method as initial inspiration, we have designed new methods with mem- ory from Steffensen’ or Traub’s schemes (in the last case we have previously transformed it in a derivative-free method), as well as a parametric family of iterative procedures of third- and fourth-order of convergence. Our statements, based on consistent discrete dynamics results and also on Feigenbaum diagrams ofthe family, allow us to select the most stable elements ofthe class and to find those that present convergence to other points different from the solution of our problem or even chaotic behavior.
There is a well known equivalence between determining multiple orthogonalpolynomials and Hermite-Pad´ e approximants of types I and II, see [9, 10, 11]. To see how to generate the multiple orthogonalpolynomialsof types I and II, and the associated polynomialsofthe second kind, see [2, 3, 4, 8, 10, 11].
The CEQS is defi ned by fi ve factors (effort, ability, preparation, persistence and unity) each consisting of four items making a total twenty items. The items are written in a clear brief way stating, «I can do» thus refl ecting the judgement ability in accordance with the recommendations by Bandura (2006). The Collective Effi cacy Questionnaire for Sports consists of an 11- point scale (0 - 10) which scores answers from «No confi dence at all» to «Absolute confi dence». The initial instructions refl ect the confi dence ofthe team’s capability when faced with the situation of competing in the near future («Grade to what extent your team believes in its abilities when faced with an imminent match or competition…»). These initial instructions are written in present tense considering that effi cacy is a changing construction not a characteristic.
lacking (Stevens et al., 2004). One ofthe most important limitations ofthe existing scales revolves around their conceptualization of leisure, which focuses onthe frequency of participation in leisure activities rather than onthe meaning and satisfaction that they lend. This is particularly the case ofthe Victoria Longitudinal Study Activity Questionnaire, a Likert-type scale-ranging from 0 (never) to 8 (daily) - which, in the adaptation by Jopp and Hertzog (2010), comprised 57 items relating to eleven different sorts of activities (physical, crafts, games, TV, social-private, social-public, religious, developmental, experiential, technology and travel). In spite of its acceptable psychometric properties, the Victoria Longitudinal Study Activity Questionnaire only provides information onthe type of leisure activity and the amount of time that the person spends doing it. Taking into account that leisure is a subjective experience (Iso-Ahola, 1980; Neulinger, 1981) which can involve different meanings, and the fact that the farther away the leisure activity is from the routine or the amount of time spent doing it are precisely the meanings that make it a potentially positive infl uence (Carbonneau, Martineau, Andre, & Dawson, 2011), it could be posited that this instrument is limited from a conceptual perspective. According to Wakui, Saito, Agree, and Kai (2012), while accepting that different activities could bring about different and specifi c effects or benefi ts, we must not overlook the different meanings that caregivers attribute to those leisure activities.
In this paper, we consider regular Borel measures µ defined on subsets ofthe complex plane which are Jordan arcs, or connected finite union of Jordan arcs, and we show how the support of µ is determined by the entries ofthe Hessenberg matrix D associated with µ. The Hessenberg matrix is the natural generalization ofthe tridiagonal Jacobi matrix to the complex plane and, in the particular case of measures with support the unit circle T, the Hessenberg matrix is a Toeplitz matrix.
Concerning dynamical models, NEGFF has been used to study the different transport material properties, such as electron conductance [33, 34] or thermoelectric characteris- tics . Generally, NEGFF is used in combination with Tight Binding (TB) approach or Density Functional Theory (DFT) in order to describe from first principles the elec- trical transport. However, the computational effort demanded by NEGFF computa- tions for systems with a large number of atoms exceeds the capabilities ofthe current high-computing facilities being unfeasible to simulate realistic devices. Thus, several approximations have to be done like decreasing the system size; considering only one or two QDs; a simplified description ofthe energy level spectra ofthe QDs or assuming constant transitions rates [36, 37, 38, 39, 40]. Although some extra implementations have been included in NEGFF, like the potential due to the self-charge [41, 42], nobody has done a fully quantum transport study in an extended arbitrary array of QDs using this framework since this approach is usually unfeasible to implement for large systems being large QD arrays a computational challenge.
Additionally, as expected, both HP and OP are positively associated with measures of valuing and passion, as in past research (Marsh et al., 2013; Vallerand et al., 2003). Surprisingly, time was only signifi cantly correlated with OP, whilst liking was only correlated with HP, instead of being both correlated with both criteria (Marsh et al., 2013; Vallerand et al., 2003, Study 1). These relationships may be better understood when we consider the type of activity separately. These results are consistent with Chamarro et al.’s (2011) fi ndings, which show that HP increases with hours of practice. It may be related to the high number of dancers involved in our sport and physical activity sample. In contrast, for videogamers, high involvement is not related to HP (Fuster et al., 2014). Also, liking was predicted by HP but not by OP. This is in line with Marsh et al. (2013), OP being associated with engagement but less with loving the activity. Nevertheless, and from the point of view of validity, the present results showed that while OP and HP were related to the various criteria, these associations varied somewhat for liking and time criteria. Future research is needed to replicate these relationship patterns with other samples.
Up to now, to the best of our information, there only exists a formal, recurrent way  to evaluate linearization coecients. Its application to the simplest case, i.e., to the expansion ofthe products of two Charlier polynomials in series of Charlier polynomialsofthe same type, leads to a six-term recursive relation forthe corresponding linearization coecients which has not yet been possible to be solved, even not at a hypergeometric level by symbolic means (Petrovsek algorithm ). Also , Dunkl  for Hahn polyno- mials and Askey and Gasper  for Kravchuk polynomials have been able to calculate explicitly the expansion coecients ofthe Clebsch-Gordan-type or conventional lineariza- tion problems (i.e., those problems which involve polynomialsofthe same system). They are collected in .
The structure ofthe paper is as follows. In Section 2 we present a survey of results surrounding the Favard Theorem when a sequence ofpolynomials satises a linear relation like (1.1). In particular, we show that the interlacing property forthe zeros of two consecutive polynomials gives basic information about the preceding ones in the sequence ofpolynomials.
el material que va a ser sometido a una serie de tratamientos previos como, por ejemplo, tratamientos térmicos de esterilización, retenga en un primer momento esos compuestos aditivados para que la liberación no se produzca en los primeros días o incluso horas, ya que para alargar la vida útil del alimento resulta más interesante que esa cesión sea lenta y progresiva. Por lo tanto, se busca una inmovilización del agente activo lo que implica una menor liberación inicial, lo que a su vez provoca que su actividad antioxidante dure un mayor tiempo. Para lograr este fin, en el último artículo recogido en esta memoria (“Art. 8. Interaction and release of catechin from anhidride maleic grafted polypropylene films”), se aborda la inmovilización de catequina en la matriz de polipropileno a través de la incorporación de polipropileno modificado superficialmente con anhídrido maleico.