The global behaviour **of** the **zeros** **of** the **discrete** **and** **continuous** classical orthogonal poly- nomials in both nite **and** asymptotic cases has received a great deal **of** attention **from** the early times [22, 27, 45] **of** approximation theory up to now [8, 9, 20, 23, 24, 31, 32, 33, 34, 36, 37, 41, 42, 46, 47, 48, 50]. Indeed, numerous interesting results have been found **from** the dif- ferent characterizations (explicit expression, weight function, **recurrence** **relation**, second order dierence or dierential equation) **of** the polynomial. See [16] for a survey **of** the published results up to 1977; more recent discoveries are collected in [49] **and** [31] for continuos **and** dis- crete **polynomials**, respectively. Still now, however, there are open problems which are very relevant by **their** own **and** because **of** its numerous applications to a great variety **of** classical systems [29, 35] as well as quantum-mechanical systems whose wavefunctions are governed by orthogonal **polynomials** in a \**discrete**" [2, 3, 40, 43, 44] or a \**continuous**" variable [5, 18, 19, 39]. In this paper the attention will be addressed to the problem **of** determination **of** the moments **of** the **distribution** density **of** **zeros** for a classical orthogonal polynomial **of** a given order n in both **discrete** **and** **continuous** cases as well as its asymptotic values (i.e, when n ! 1 ), which

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The classical orthogonal **polynomials** are very interesting mathematical objects that have attracted the attention not only **of** mathematicians since **their** appearance at the end **of** the 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. [1]). Among such properties, a fundamental role is played by the so-called characterization theorems, i.e., such properties that completely deﬁne **and** characterize the classical **polynomials**. Obviously not every property characterize the classical **polynomials** **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 [7])—, the TTRR characterizes the orthogonal **polynomials** (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 for the classical **discrete** [3,6] **and** the q-classical [4,10] **polynomials**. For the **continuous** case see [8,9].

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The method **of** proof used is very straightforward. It is based on an explicit formula for the moments-around-the-origin **of** the **discrete** density **of** **zeros** **of** a polynomial with a given degree in terms **of** the coecients **of** the three term **recurrence** **relation** [37], as described in Lemma 1 given below. This method was previously employed to normal (non-q) **polynomials** where **recurrence** coecients are given by means **of** a rational function **of** the degree [38], as well as to correspond- ing Jacobi matrices [39] encountered in quantum mechanical description **of** some physical systems. The paper is structured as follows. Firstly, in section 2, one introduces a general set **of** q- **polynomials** f P n ( x ) q g Nn

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Markett [37] for symmetric orthogonal **polynomials**, has designed a method which be- gins with the three-term **recurrence** **relation** **of** the involved orthogonal polynomial system to set up a partial dierence equation for the (orthogonal) polynomial, in case **of** connec- tion problems, or for the product **of** two (orthogonal) **polynomials**, in case **of** linearization problems, to be expanded; then, this equation has to be solved in terms **of** the initial data. Ronveaux et al. [47, 48] for classical **and** semiclassical orthogonal **polynomials** **and** Lewanowicz [30, 34] for classical orthogonal **polynomials** have proposed alternative, sim- pler techniques **of** the same type although they require the knowledge **of** not only the **recurrence** **relation** but also the dierential-dierence **relation** or/**and** the second-order dierence equation, respectively, satised by the **polynomials** **of** the orthogonal set **of** the expansion problem in consideration. See also [5, 11, 23, 31, 45, 46] for further description **and** applications **of** this method in the **discrete** case, **and** [19, 32] in the **continuous** case as well as [3, 33] for the q -**discrete** orthogonality. Koepf **and** Schmersau [25] has proposed a computer-algebra-based method which, starting **from** the second order dierence hyper- geometric equation, produces by symbolic means **and** in a recurrent way the expansion coecients **of** the classical **discrete** orthogonal hypergeometric **polynomials** (CDOHP) in terms **of** the falling factorial **polynomials** (already obtained analytically by Lesky [28]; see also [41], [42]) as well as the expansion coecients **of** its corresponding inverse problem. The combination **of** these two simple expansion problems allows these authors to solve the connection problems within each specic CDOHP set.

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The **recurrence** relations for the special functions (**and** thus for the associated wave functions) are interesting not only **from** the theoretical point **of** view (they are useful for computing the values **of** matrix elements **of** certain physical quantities see e.g. [11] **and** references therein) but also they can be used to numerically compute the values **of** the functions as well as **their** derivatives, as it is shown in [6] for the case **of** Laguerre **polynomials** **and** the associated wave functions **of** the harmonic oscillator **and** the hydrogen atom. Nevertheless, we need to point out that, although the **recurrence** relations seem to be more useful for the evaluation **of** the corresponding functions than other direct methods, one should be very careful when using them (see e.g. the nice surveys [16, 23] on numerical evaluation **and** the convergence problem that appears when dealing with **recurrence** relations, or the most recent ones [17, 18] for the case **of** the hypergeometric function 2 F 1 ).

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The behavior **of** these fixed points is reflected in the dynamical planes; it can be observed that the roots **of** the polynomial **and** also the possible attracting fixed points are located on the bisectrix z = x **of** the plane. The stable behavior **of** the origin can be seen at Figure 5b, where the basin **of** attraction **of** point (0, 0) is plotted in blue. Moreover, in Figures 5c, 5d **and** 5e only the basins **of** attraction **of** the solutions are found but the second one shows wider regions **of** stability; let us remark that this picture corresponds to the value **of** θ that provides higher order **of** convergence **of** the family. Finally, a highly unstable behavior can be observed in Figures 5a **and** 5f, where big black areas **of** no convergence to the roots appear **and** the basins **of** attraction **of** the roots are very narrow.

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According to Dr.Marshall Chasin **from** the hearing journal, a critical bandwidth is approximately a third **of** an octave, which is why one-third octave bandwidth analysis is so common in our field. One can attribute loudness for a single pure tone at 1000 Hz. Adding another at 1050 Hz will slightly increase its intensity but not its loudness. The same is true **of** adding a third pure tone at 1100 Hz, **and** this holds true as long as the sound energy is within a critical bandwidth. When adding another pure tone at 1200 Hz (exceeding the critical bandwidth at 1000 Hz, which is about 160 Hz), however, a person will subjectively report that the sound is now louder because the sound spectrum has exceeded the critical bandwidth. A bandwidth is explained as well using filters, for example a band-pass filter allows a range **of** frequencies within the bandwidth to pass through while stopping those outside the cut-off frequencies as it is shown Figure 2.

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The research **of** the rst author was supported by a grant **of** Instituto de Cooperacion Iberoamericana (I.C.I.) **of** Spain. He is very greateful to Departamento de Ingeniera, Uni- versidad Carlos III de Madrid for his kind hospitality. The research **of** the second author was supported by Comision Interministerial de Ciencia y Tecnologa (CICYT) **of** Spain under grant PB 93-0228-C02-01.

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Palfrey 2002 **and** Hummel, 2010). That is, we consider that two …rms compete in catalogs (locations **and** prices at the same time) but, compared to the standard setup in which …rms locate at a point in the unit interval we focus on a case in which …rms can locate to the western, to the central or to the eastern district **of** the linear city. We have to stress here, though, that despite the fact that the set **of** locations in our setup is …nite, the set **of** available prices is not. Therefore, since the strategy space **of** each …rm is the Cartesian product **of** these two sets, we have that the strategy space **of** each …rm is in…nite. This, along with the facts that the game is not a constant-sum game **and** that …rms’ payo¤ functions exhibit discontinuities, implies that existence **of** a unique symmetric equilibrium **and**, thereafter, possibility **of** full characterization are not guaranteed by any known theorem.

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are valid (Fitzpatrick **and** Hargrove 2009). Our results suggest that, similarly to spatial extrapolation (Heikki- nen et al. 2012), a good capability **of** SDMs to predict species distributions under training conditions does not guarantee equally good performance when these are transferred in time. In spite **of** this, environmental extrapolation seems to be a situation that often cannot be avoided when correlative SDMs are being transferred in space or time. For this reason, our findings demon- strate the importance **of** environmental extrapolation for temporal transference **of** SDMs, **and** is consistent with the recommendations **of** reporting the degree **of** environ- mental extrapolation both for temporal **and** spatial transference **of** SDMs (e.g., Elith et al. 2010, Zurell et al. 2012, Mesgaran et al. 2014) to prevent erroneous or imprecise predictions, or at least communicate where model predictions are reliable **and** where they are not. Bayesian Hierarchical models (e.g., Dynamic Range Models; Pagel **and** Schurr 2009) have been shown to pro- duce reliable predictions in time (using simulated data; Schurr 2012), with several advantages including the inference **of** spatiotemporal range dynamics in equilib- rium **and** non-equilibrium conditions, **and** the capacity **of** reporting the within-model uncertainty **of** the predic- tions. These advantages could be especially relevant in predicting range dynamics when models are transferred in time or space **and** extrapolated to novel environmen- tal conditions (Schurr 2012).

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007 Price, K., M. Bussing, W.A. Bussing, D. Maurer & C. Bartels 1980. Finfish survey: 83-144. In: D, Mau- rer, C. Epifanio **and** K. Price (Eds.). Ecological as- sessment **of** finfish **and** megabenthic invertebrates as indicators **of** natural **and** impacted habitats in the Gulf **of** Nicoya, Costa Rica. Progress Report **of** the 1979 International Sea Grant Program University **of** Delaware, College **of** Marine Studies, Newark. 008 Maurer, D., C. Epifanio, K. Price, J.A. Vargas, M.M.

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[7] M. Cotlar **and** L. Recht, Rodolfo A. Ricabarra (August 1925–November 1984), in Volume in homage to Dr. Rodolfo A. Ricabarra (Spanish), i–v, Univ. Nac. del Sur, Bah´ıa Blanca, 1995. [8] M. Cotlar **and** C. Sadosky, The Adamjan-Arov-Kre˘ın theorem in general **and** regular repre- sentations **of** R 2 **and** the symplectic plane, in Toeplitz operators **and** related topics (Santa

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claims “We have been in possession **of** this proof for several years. Recently Favard published an identical proof in the Comptes Rendus”. Also Natanson in his book [17, p. 167] said “This theorem was also discovered (independent **of** Favard) by the author (Natanson) in the year 1935 **and** was presented by him in a seminar led by S.N. Bernstein. He then did not publish the result since the work **of** Favard appeared in the meantime”. The “same” theorem was also obtained by Perron [19], Wintner [28] **and** Sherman [20], among others.

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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**.

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Abstract. A distributional analysis **of** 81 gymnosperm genera was undertaken. On the basis **of** the congruence in the **distribution** **of** these genera, nine areas **of** endemism were recognised. Many **of** these areas also represent areas **of** endemism for other plant **and** animal taxa. South-western China **and** New Caledonia are particularly interesting **from** the viewpoint **of** gymnosperm diversity **and** endemism. The suggested areas **of** endemism agree in part with some floristic regions previously proposed. The congruence between the areas **of** endemism suggested **and** postulated Pleistocene refuges **and** panbiogeographic nodes is discussed. A cladistic biogeographic analysis was carried out **and** a general area cladogram obtained by strict consensus shows two major components, one Gondwanic **and** the other almost Laurasian. This cladogram was compared with previous studies **and** the similarities **and** differences among relationship areas are discussed.

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where the apostrophe 0 denotes matrix transpose. We emphasize that the con- struction **of** a practical optimization algorithm **from** Hopfield networks stems **from** the dynamical properties [2]: since a Lyapunov function is decreasing over time, thus the evolution **of** the states **of** the network heads towards a stable equilibrium, where the Lyapunov function presents a—possibly local—minimum. Therefore, an optimization problem where the target function has the same struc- ture **of** the Lyapunov function given by Equation (6) is solved by matching the target **and** the Lyapunov function, so that the weights **and** biases are obtained. Finally, the network is “constructed”, which usually means that some numerical method is used to integrate Equation (4) until the states reach an equilibrium, which provides the minimum. Therefore, using a numerical method that pre- serves the dynamical properties **of** the **continuous** ODE under discretization is a crucial step, since a solution is only obtained by an algorithm that mimics the correct dynamical behaviour.

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After this paper was flnished, the authors knew that the same result had been obtained independently **and** at the same time by Ponnaian **and** Shanmugam in [8]. Although both papers use Theorem 3, the way those authors follow to study the **zeros** **of** the G function is different **from** ours. Ponnaian **and** Shanmugam base **their** arguments on the study that they carry out about the roots **of** the exponential Euler splines. In particular, they need to obtain a recursion **relation** for those exponential splines. Moreover, they consider four different cases on the degree d. On the contrary, our arguments may be more direct, since we apply already known properties **of** the Euler-Frobenius **polynomials** **and** only have to consider even **and** odd degree cases separately.

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In other words, these **polynomials** are obtained via the addition **of** delta Dirac measures to a positive denite weight function [11, 14]. A special emphasis was given to the case when is the classical \**discrete**" weight function corresponding to the classical \**discrete**" [12] **polynomials** **of** Hahn [3], Meixner [1, 2, 6], Kravchuk [2] **and** Charlier [2, 7] for which the coecients , **and** in (1.2) are **polynomials** such that **and** do not depend **of** n , is a constant, degree( ) 2 **and** degree( ) 1. In all these cases, when one or

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The rst approach to such a kind **of** modications **of** a moment linear functional is [3]. In particular, necessary **and** sucient conditions for the existence **of** a sequence **of** **polynomials** orthogonal with respect to such a linear functional are obtained. Furthermore, an extensive study for the new orthogonal **polynomials** was performed when the initial functional is semi- classical.

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Special cases **of** quasi-definite linear functionals are the classical ones (those **of** Ja- cobi, Laguerre, Hermite, **and** Bessel). In the last years perturbations **of** the func- tional via the addition **of** Dirac delta functions —the so-called Krall-type orthog- onal **polynomials**— have been extensively studied (see e.g. [6, 7, 16, 20, 21, 22, 26] **and** references therein), i.e., the linear functional

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