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159
QuasiOrthogonality With Applications to Some Families of Classical Orthogonal Polynomials
, 2002
"... In this paper, we study the quasi{orthogonality of orthogonal polynomials. New results on the location of their zeros are given in two particular cases. Then these results are applied to Gegenbauer, Jacobi and Laguerre polynomials when the restrictions on the parameters involved in their de niti ..."
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Cited by 13 (1 self)
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In this paper, we study the quasi{orthogonality of orthogonal polynomials. New results on the location of their zeros are given in two particular cases. Then these results are applied to Gegenbauer, Jacobi and Laguerre polynomials when the restrictions on the parameters involved in their de nitions are not satis ed. The corresponding weight functions are investigated and the location of their zeros is discussed.
Closed forms: what they are and why we care
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not cl ..."
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Cited by 13 (4 self)
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The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.
2000), Numerical and asymptotic aspects of parabolic cylinder functions
 J. Comp. Appl. Math
"... Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are mo ..."
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Cited by 12 (8 self)
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Several uniform asymptotics expansions of the Weber parabolic cylinder functions are considered, one group in terms of elementary functions, another group in terms of Airy functions. Starting point for the discussion are asymptotic expansions given earlier by F.W.J. Olver. Some of his results are modified to improve the asymptotic properties and to enlarge the intervals for using the expansions in numerical algorithms. Olver’s results are obtained from the differential equation of the parabolic cylinder functions; we mention how modified expansions can be obtained from integral representations. Numerical tests are given for three expansions in terms of elementary functions. In this paper only real values of the parameters will be considered. 1991 Mathematics Subject Classification: 33C15, 41A60, 65D20.
Allpay auctions with endogenous rewards
, 2009
"... This paper examines a perfectly discriminating contest (allpay auction) with two asymmetric players. Valuations are endogenous and depend on the effort each player invests in the contest. The shape of the valuation function is common knowledge and differs between the contestants. Some key propertie ..."
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Cited by 12 (1 self)
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This paper examines a perfectly discriminating contest (allpay auction) with two asymmetric players. Valuations are endogenous and depend on the effort each player invests in the contest. The shape of the valuation function is common knowledge and differs between the contestants. Some key properties of R&D races, lobbying activity and sport contests are captured by this framework. Once the unique equilibrium in mixed strategies analyzed, we derive a closed form of the expected expenditure of both players. We characterize the expected expenditure by the means of incomplete Beta functions. We focus on unordered valuations.
Large parameter cases of the Gauss hypergeometric function
, 2002
"... We consider the asymptotic behaviour of the Gauss hypergeometric function when several of the parameters a, b, c are large. We indicate which cases are of interest for orthogonal polynomials (Jacobi, but also Meixner, Krawtchouk, etc.), which results are already available and which cases need more a ..."
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Cited by 12 (1 self)
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We consider the asymptotic behaviour of the Gauss hypergeometric function when several of the parameters a, b, c are large. We indicate which cases are of interest for orthogonal polynomials (Jacobi, but also Meixner, Krawtchouk, etc.), which results are already available and which cases need more attention. We also consider a few examples of 3 F2 functions of unit argument, to explain which difficulties arise in these cases, when standard integrals or differential equations are not available.
8 Lectures on Quantum Groups and qSpecial Functions
, 1996
"... The lecture notes contains an introduction to quantum groups, qspecial functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the AskeyWilson polynomials out in detail ..."
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Cited by 12 (2 self)
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The lecture notes contains an introduction to quantum groups, qspecial functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the AskeyWilson polynomials out in detail as the main example. As an application we derive an addition formula for a twoparameter subfamily of AskeyWilson polynomials. A relation between the AlSalam and Chihara polynomials and the quantised universal enveloping algebra for su(1, 1) is given. Finally, more examples and other approaches as well as some open problems are given.
Hermite Polynomials in Asymptotic Representations of Generalized Bernoulli, Euler, Bessel, and Buchholz Polynomials
, 1999
"... This is the second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli, Euler, Be ..."
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Cited by 10 (3 self)
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This is the second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli, Euler, Bessel and Buchholz polynomials. The asymptotic approximations of these polynomials are valid for large values of a certain parameter. The representations and limits include information on the zero distribution of the polynomials. Graphs are given that indicate the accuracy of the first term approximations.
On Polynomials Related with HermitePadé Approximations to the Exponential Function
"... We investigate the polynomials P n ; Qm and R s , having degrees n; m and s respectively, with P n monic, that solve the approximation problem Enms (x) := P n (x)e \Gamma2x +Qm (x)e \Gammax +R s (x) = O(x n+m+s+2 ) as x ! 0: We give a connection between the coefficients of each of the polynomi ..."
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Cited by 9 (2 self)
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We investigate the polynomials P n ; Qm and R s , having degrees n; m and s respectively, with P n monic, that solve the approximation problem Enms (x) := P n (x)e \Gamma2x +Qm (x)e \Gammax +R s (x) = O(x n+m+s+2 ) as x ! 0: We give a connection between the coefficients of each of the polynomials P n ; Qm and R s and certain hypergeometric functions, which leads to a simple expression for Qm in the case n = s. The approximate location of the zeros of Qm , when n AE m and n = s, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations of P n ; Qm ; R s and Enms are given and, using saddle point methods, we derive the exact asymptotics of P n ; Qm and R s as n; m and s tend to infinity through certain ray sequences. We also discuss aspects of the more complicated uniform asymptotic methods for obtaining insight into the zero distribution of the polynomials, and we give an example showing the zeros of the polynomials P n ; Qm and R s for the case n = s = 40; m = 45. 1991 Mathematics Subject Classification: 41A21, 30E15, 30C15, 41A60. Keywords & Phrases: HermitePad'e approximant, exponential function, hypergeometric series, error asymptotics, zero distribution, HermitePad'e Type I polynomials. Note: Work carried out under project MAS2.8 Exploratory research. 1.
Computing complex Airy functions by numerical quadrature
 Numer. Algorithms
"... CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms. ..."
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Cited by 8 (5 self)
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CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics. CWI's research has a themeoriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.