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 thedistributional 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-polynomials in terms of the coeMcients of thepolynomials and of thedistributional 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.
These two mechanisms will have a final net effect on disposable income, which will have effects on welfare through changes in consumption. In order to quantify the welfare effect of the different fiscal policies, I use a measure of the consumption equivalent variation. I quantify the welfare effect of a given policy framework for an individual by asking: by how much the consumption has to change in all future periods and in the initial steady state, so that the expected utility equals the one after the transition, under a specific policy frame- work? In other words, by how much, in consumption terms, the agents benefit or lose from a specific fiscal policy framework in an economic downturn context? In order to studythedistributional effects, I compute the Gini coefficient in the initial steady state, and I compare it with the Gini coefficients that result from the transition of each policy framework.
lattices. Onthe other hand, using theq-analog of the quantum theory of angular momentum [20–23] we can obtain several results for theq-polynomials, some of which are nontrivial from the viewpoint of the theory of orthogonal polynomials (see, e.g., the nice surveys [24, 25]). In fact, in this paper we present a detailed study of some q-analogs of the Racah polynomials u α,β n (x(s), a, b) q and u e
In this paper we study in detail a q-extension of the generalized Hermite polynomials of Szeg˝ o. A continuous orthogonality property on R with respect to the positive weight function is established, a q-difference equation and a three-term recurrence relation are derived for this family of q-polynomials.
Let us also point out that there are also the so-called discrete (see e.g. ) and q-discrete Sobolev type orthogonal polynomials associated with the classical discrete and q-classical functionals [23, 24]. In both cases the corresponding polynomials can be reduced to the Krall-type one (except for theq-case when the mass is added at zero where a more careful study is needed [23, 24]) since the differences ∆f (x) = f (x + 1) − f(x) and D q f(x) = (f (qx) − f(x))/(qx − x).
In 1940, H. L. Krall  obtained three new classes of polynomials orthogonal with respect to measures which are not absolutely continuous with respect to the Lebesgue measure. In fact, his study is related to an extension of the very well known characteri- zation of classical orthogonal polynomials by S. Bochner. This kind of measures was not considered in . Moreover, in his paper H. L. Krall obtain that these three new families of orthogonal polynomials satisfy a fourth order dierential equation. The corresponding measures are given in the following table.
User acceptability has become a critical issue for the successful imple- mentation of transport pricing measures and policies. Although several studies have addressed the public acceptability of road pricing, little evidence can be found of the effects of pricing strategies. The accept- ability of alternative schemes for a toll network already in operation is an issue to be tackled. This paper contributes to the limited literature in this ﬁeld by exploring perceptions toward road-pricing schemes among toll road users. Onthe basis of a nationwide survey of toll road users in Spain, thestudy developed several binomial logit models to analyze user acceptability of three approaches: express toll lanes, a time-based pricing approach, and a ﬂat fee (vignette) system. The results show notable differences in user acceptability by the type of charging scheme proposed. Express toll lanes were more acceptable by travelers who perceived greater beneﬁts from saving travel time. The acceptability of time-based approaches (peak versus off-peak) decreased for users who felt forced to use the toll road, whereas this was not an aspect that signiﬁcantly inﬂuenced users’ support for ﬂat fee schemes. In addi- tion, a ﬂat fee strategy was more acceptable for long-distance trips and truck drivers who regularly used the toll facilities. The results from this analysis can inform policy makers and planners for the promotion of more efﬁcient, socially inclusive, and publicly acceptable road-pricing schemes.
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 the last decade an increasing interest onthe so called q-orthogonal polynomials (or basic orthogonal polynomials) is observed ( for a review see ,  and ). The reason is not only of purely intrinsic nature but also because of the so many applications in several areas of Math- ematics ( e.g., continued fractions, eulerian series, theta functions, elliptic functions,...; see for instance  and ) and Physics ( e.g., angular momentum  and  and its q-analog -, q-Shrodinger equation  and q-harmonic oscillators -). Moreover, it is well known the connection between the representation theory of quantum algebras (Clebsch-Gordan coecients, 3j and 6j symbols) and theq-orthogonal polynomials, (see ,  (Vol. III), , ,  ), and the important role that these q-algebras play in physical applications (see for instance - and references therein).
The symmetries of quantum states play an important role in explaining the degeneracy of energy levels . The connection of the energy levels of the hydrogen atom with the irreducible representation of O(4, 2) conformal symmetry was found in . Theq-deformed Heisenberg–Weyl symmetry was used to introduce the notion of quantum q-oscillators [3, 4]. The physical meaning of q-deformations was clarified in  where it was shown that classical q-oscillators and their quantum partners are standard nonlinear oscillators vibrating with a frequency depending onthe amplitude. Thus, the symmetry groups and theq-deformed symmetry groups are important ingredients in the description of states in quantum optics and quantum mechanics. A general consideration of constructing the irreducible representations of Lie groups and their connection with the formalism of classical mechanics was presented in  in the context of symmetry applications in quantum mechanics and quantum optics.
It is well known that the Lie Groups Representation Theory plays a very important role in the Quantum Theory and in the Special Function Theory. The group theory is an ef- fective tool for the investigation of the properties of dierent special functions, moreover, it gives the possibility to unify various special functions systematically. In a very simple and clear way, onthe basis of group representation theory concepts, the Special Function Theory was developed in the classical book of N.Ya.Vilenkin  and in the monography of N.Ya.Vilenkin and A.U.Klimyk , which have an encyclopedic character.
tion of the non-standard q-Racah polynomials was considered in detail. This is an important example for two reasons: 1) it is the first family of the Krall- type polynomialson a non-linear type lattice that has been studied in detail and 2) almost all modifications (via the addition of delta Dirac masses) of the classical and q-classical polynomials can be obtained from them by taking appropriate limits (as it is shown for the dual q-Hahn, the Racah, and theq- Hahn polynomials in section 4.2). Let us also mention here that an instance of the Krall-type polynomials obtained from the Askey-Wilson polynomials (with a certain choice of parameters), by adding two mass points at the end of the orthogonality has been mentioned in [20, §6, page 330]. This Askey-Wilson-Krall-type polynomials solve the so-called bi-spectral problem associated with the Askey-Wilson operator. Then, it is an interesting open problem to studythe general Krall-type Askey-Wilson polynomials and to obtain their main properties. This will be considered in a forthcoming paper. Acknowledgements:
Mellin integral transforms for some families of basic hypergeometric polynomials from the Askey scheme  were considered in .Derivation of these Mellin transform pairs is essentially based onthe use of Ramanujan’s q-extension [17,4,5] of the Euler integral representation for the gamma function
In the ﬁrst section we consider the Aron-Berner extension. We begin with the Arens product in a commutative Banach algebra, a very speciﬁc extension of a bilinear function, but nevertheless an extension in which some of the main points of more general extensions already appear, such as lack of symmetry and the notion of regularity. We then deﬁne and studythe Aron-Berner extension, an extension of polynomials from a Banach space to its bidual. In the second section, we consider extensions from E to G. Here all solutions stem from the existence of a continuous linear extension morphism for linear forms E −→ G , a condition obviously stronger than Hahn-Banach, and not satisﬁed in all cases. Section 3 is devoted to Hahn-Banach type extensions. We are naturally drawn to ‘linearization’ of polynomials, and thus to preduals of spaces of polynomials. The space of ‘extendible’ polynomials is considered also.
(a small subset of theq-world). Here we need to point out that exits two dierent point of view in thestudy of theq-polynomials. The rst one, in the framework of theq-basic hypergeometric series , ,  and the second, in the framework of the theory of dierence equations developed by Nikiforov et al. , , . In this work we will use the second one because it gives us the possibility to provide an uniform treatment of several classes of orthogonal polynomials and, probably, it is the best way to nd further applications.
lae, epigastric lobes semicircular, anteriorly not well delimited, frontal sur- face flat, postgastric pits present, branchio-urogastric, branchio-cardiac, and branchio-intestinal grooves narrow, urogastric groove not demarcated; ab- dominal segments III to VI fused, but suture 3/4 still visible in males, abdo- men triangular, outer margins slightly concave, telson triangular, with outer margin slightly sinuous (Fig. 18 G); first pereiopods heterochelous; merus of chela in large males with distal spine on external margin, carpus with promi- nent, acute, distal spine on internal margin, palm with small, spine on up- per margin near articulation of dactylus, fingers elongated (1.6 times the length of palm) with longitudinal ridges, not gaping when closed (Fig. 18 I). First male gonopod tapering to apex, recurved laterally; mesial side con- cave; lateral side slightly convex; margin regularly displaced to mesocephalic surface and emerging on caudal surface near apex; lateral lobe absent (Fig. 18 A – C); apex outline V-shape, opening below corneous prominent spine- like process (Fig. 18 E); apical translucent spines, decreasing in size distally, forming continuous patch over caudal and lateral surfaces, mesial side with rows of translucent spines, basal lateral side with conspicuous long setae (Fig. 18 A – E). Second gonopod considerable longer than first; recurved mesially in form of question mark (Fig. 18 F).