Abstract. In the scalar case, it is well known that the zonal spherical func- tions of any compact Riemannian symmetric space of rank one can be ex- pressed in terms of the Jacobi **polynomials**. The main purpose of this paper is to revisit the matrix valued spherical **functions** associated to the complex projective plane to exhibit the interplay among these **functions**, the matrix hypergeometric **functions** **and** the matrix **orthogonal** **polynomials**. We also obtain very explicit expressions for the entries of the spherical **functions** in the case of 2 × 2 matrices **and** exhibit a natural sequence of matrix **orthogonal** **polynomials**, beyond the group parameters.

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For any realistic composite quantum-mechanical system 共 atoms, molecules, nuclei, etc. 兲 the treatment of the continu- ous part of the spectrum is a difficult task. This is especially so in the case of weakly bound systems, when both bound **and** unbound states have to be treated on equal footing. The continuum wave **functions** depend on a continuously varying parameter 共 the energy or the wave number 兲 **and** are not nor- malizable, which makes them awkward for actual applica- tions. Nevertheless, in some cases the exact non- normalizable continuum wave **functions** can be explicitly used in the calculation. This is the case for the evaluation of excitation **functions** for an operator that connects a bound state with the continuum states of a system. In this situation, the bound character of the state allows for an explicit evalu- ation of the matrix elements. This is also the case in reaction calculations in a distorted-wave Born approximation ap- proach. The transition amplitudes can be calculated from the matrix element of the relevant interaction between the initial bound state **and** the final unbound state.

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A detailed study of this family was done in [16] **and** their main characteristics are given in Table 2 of [16]. Let us now study the duality properties of the 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 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.

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but also other useful properties: they are the eigenfunction of a second order linear differential operator with polynomial coefﬁcients, their derivatives also constitute an **orthogonal** family, their generating **functions** can be given explicitly, among others (see for instances [1,8,24,25] or the more recent work [3]). Among all these properties there are very important ones that characterize these families of **polynomials**.

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The structure of the paper is as follows. In Section 2, we provide the basic properties of the classical **orthogonal** **polynomials** of discrete variable which will be needed, as well as the main data for the Meixner, Kravchuk **and** Charlier **polynomials**. In Section 3 we deduce expressions of the 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) of the classical **orthogonal** **polynomials** we nd the TTRR which satisfy the perturbed ones. In Section 6, from the relation of the perturbed **polynomials** P An (x) as a linear combination of the classical ones, we nd the tridiagonal matrices associated with the perturbed monic **orthogonal** polinomial sequence (PMOPS) f P An (x) g

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The structure of the paper is as follows. In Section 2 we list some of the main properties of the classical 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 of the generalized Bessel **polynomials** in terms of the 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.

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Our aim here is to express in terms of higher order hypergeometric Lauricella functions the corresponding asymptotic contracted measure of zeros for the sequence {P n (x)}™ =1 to be de[r]

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There are many instances, however, when the Hessenberg matrix can not computed completely, but only finite sections of it, **and** it is not possible to compute the limits of the diagonals of D. In this case, it is still possible to compute approximations of the support of the measure µ obtained computing the image of the unit circle under suitable approximations of the Riemann map. Specifically, since the coefficients of the Riemann map are the limits of the elements in each of the diagonals of the Hessenberg matrix, we may consider, as approximations of the Riemann map φ, the **functions**

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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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[8] R. Koekoek **and** R. F. Swarttouw The Askey-scheme of hypergeometric orthog- onal **polynomials** **and** its q-analogue. Reports of the Faculty of Technical Math- ematics **and** Informatics No. 94-05. Delft University of Technology. Delft 1994. [9] T.H. Koornwinder: Compact quantum groups **and** q-special **functions**. In Rep-

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addition of one or more delta Dirac **functions**. Some examples studied by dierent authors are considered from an unique point of view. Also some properties of the Krall **polynomials** are studied. The three-term recurrence relation is calculated explicitly, as well as some asymptotic formulas. With special emphasis will be considered the second order dierential equations that such **polynomials** satisfy which allows us to obtain the central moments **and** the WKB approximation of the distribution of zeros. Some examples coming from quadratic transformation polynomial mappings **and** tridiagonal periodic matrices are also studied.

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At present, interest in the study of multiple **orthogonal** **polynomials** has in- creased due to the development of simultaneous rational approximation of **functions** **and** the link between the two theories. **Orthogonal** **polynomials** **and** Pad´ e approximants are essential areas for research, because of its applications in different branches of mathematics such as number theory, the problem of moments, the analytic extension, interpolation problems, spectral theory of operators **and** others.

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of such properties in terms of the location of the mass points with respect to the support of the measure. Particular emphasis was given to measures supported in [ 1 ; 1] **and** satisfying some extra conditions in terms of the parameters of the three term recurrence relation which the corresponding sequence of **orthogonal** **polynomials** satisfy.

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Confluent hypergeometric **functions** are solutions of the confluent hypergeometric equa- tion, a modification of the Gaussian hypergeometric equation by what is called as the confluence of two of its singularities. We will show how to resolve this equation, show its integral representations, some of its properties, **and** then explore how the Confluent Hypergeometric equation can be reduced to our two cases of interest: The Weber **and** the Bessel equations. The description of these **functions** **and** the notation are according to [3], which allowed us to prove the First Kummer Formula **and** perform the connection formulae for the Weber **functions** that will be studied in Chapter 7.

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For expository reasons, we have chosen to structure this paper differently from the way we have presented the introduction, starting, first (see Section 2), by the notion of α-analytic multivariate **polynomials** **and** studying their basic algebraic properties; in particular those concerning factorization, gcd’s **and** resultants. Then, in the last Section 3, we present the generalization of CR conditions to this new setting **and** we show that they (the new conditions) characterize components of α-analytic multivariate **polynomials** (cf. Theorem 20). Moreover, as in the classical Complex Variables context, we can deduce again, from this non-standard CR conditions, some important properties of analytic **polynomials** (cf. Theorem 22).

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Abstract: On this paper a research about the submodular **functions** is made, the basis of the work comes from the studies made by Satoru Fujishije [ 7 ]. The polymatroid **and** submodular system concepts are studied exhibiting together some examples. These concepts are applied to the construction of the Greedy Algorithm, which can solve certain type of problems of linear optimization.

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roots. However further research is needed to explain the different patterns observed across tissues **and** meadows. Moreover, this work represents the first identification of endophytic bacterial present in P. oceanica tissues, with very suggestive results. Some of the sequences were closely related to major groups of bacteria able to fix nitrogen, some others related to the sulfur cycle **and** finally a group of sequences had their closest known relatives among those found in corals affected by black band disease. It is not possible to infer whether or not the functional genes **and** capacities associated to the closest matching relatives will be present in our samples, due to the low similarity of some sequences to known cultured bacteria or even to environmental sequences. However, the fact that the closest matches are related to these three categories suggests that endophytic bacteria may play an important role in the health of P. oceanica, by providing nitrogen or protecting the plants against the invasion of toxic sulfides, whereas some others may be pathogenic. Moreover, the low sequence similarity to previously reported sequences in Genbank indicates that many of these sequences correspond to unknown bacteria some of which could be specific of P. oceanica tissues. Subsequent research should include a search for functional genes involved in nitrogen fixation **and** the Sulfur cycle **and** also a more detailed study on healthy vs. damaged tissues of P. oceanica, which could lead to the discover of new pathogens of marine angiosperms.

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5 Williams (1966: 6) sostiene que “We attribute the origin **and** perfection of this design to a long period of selection for effectiveness in this particular role.” La dimensión histórica es patente, sólo podemos decir de un rasgo que es una adaptación si ese rasgo ha sido seleccionado en virtud de la funcionalidad que entraña y de la eficacia con que dota al organismo durante un largo periodo temporal. 6 Waddington experimentó los efectos del éter sobre las huevas de Drosophila. Para ello sometió a las huevas a una dosis subletal de vapores de éter y observó que mientras muchos de los supervivientes nacían normalmente, otros pocos desarrollaban un bitórax. Estos ejemplares anormales fueron seleccionados como progenitores de la siguiente camada y las huevas que produjeron fueron igualmente sometidas a la misma exposición experimental de vapores de éter observándose resultados análogos. Sin embargo, a partir de una determinada generación la aparición del bitórax es generada sin necesidad de exposición alguna a los vapores de éter. Waddington denomina a este proceso asimilación genética y señala que el proceso de selección natural debe ser suplementado por la asimilación genética en el desarrollo de las características fenotípicas. Waddington no es lamarckista, pese a lo que imprudentemente pudiera pensarse acerca de su dispositivo experimental. La idea que quiere poner de manifiesto es que el genotipo es canalizado por las condiciones ambientales, dicho de otra manera, el genotipo asimila las condiciones ambientales y responde en consecuencia con un fenotipo adap- tado. Si se desarrolla un bitórax no es de la nada, sino porque existen alelos generadores de bitórax en el genotipo de Drosophila. La presencia o no de éter modifica la expresión de estos alelos y, en consecuencia, afecta al incremento de su frecuencia en las poblaciones experimentales de Drosophila.

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Orbitofrontal syndrome has been associated with disin- hibition, inappropriate behaviors, irritability, mood lability, tactlessness, distractibility, **and** loss of import to events. Affect may become extreme with moria (an excited affect) or Witzelsucht (the verbal reiteration of caustic or facetious remarks), fi rst noted by Oppenheim (1890, 1891). Individuals with this syndrome are unable to respond to social cues, **and** they are stimulus bound. Cummings (1993) noted that automatic imitation of the gestures of others may occur with large lesions. Interestingly, it has been noted that these patients have no diffi culty with card-sorting tasks ( Laiacona et al., 1989). Eslinger **and** Damasio (1985) coined the term “acquired sociopathy” to describe dysregulation that couples both a lack of insight **and** remorse regarding these behaviors. Much of this may refl ect the stimulus- bound nature of this disorder. The orbitofrontal cortex appears to be linked predominantly with limbic **and** basal forebrain sites. The orbital prefrontal cortex may have the ability to maintain its own level of functional arousal due to its cholinergic innervation from the basal forebrain ( Mesulam , 1986). According to Fuster (2002), the ventromedial areas of the prefrontal cortex are involved in expression **and** control of emotional **and** instinctual behaviors.

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Uvarov, Polynomial Solutions of hypergeometri type dierene Equations. and their lassiation[r]

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