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66
Computing Special Functions By Using Quadrature Rules
, 2002
"... The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, di#erential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles f ..."
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The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, di#erential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles from asymptotic analysis, we develop reliable algorithms which are valid for large domains of real or complex parameters. Our present investigations include Airy functions, Bessel functions and parabolic cylinder functions. In the case of Airy functions we have improvements in both accuracy and speed for some parts of Amos's code for Bessel functions. 2000 Mathematics Subject Classification: 65D20, 65D32, 33C10, 33F05, 41A60.
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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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.
Numerical Algorithms for Uniform Airytype Asymptotic Expansions
 CENTRUM VOOR WISKUNDE EN INFORMATICA
, 1997
"... Airytype asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic ..."
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Airytype asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we don't need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.
The DLMF Project: A New Initiative in Classical Special Functions
 International Workshop on Special Functions  Asymptotics, Harmonic Analysis and Mathematical Physics. Hong Kong
, 2000
"... that aims to produce a successor to Abramowitz and Stegun’s Handbook of Mathematical Functions, published by the National Bureau of Standards in 1964 and reprinted by Dover in 1965. Both editions continue to sell briskly and are widely cited in the scientific literature. However, with the many advan ..."
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that aims to produce a successor to Abramowitz and Stegun’s Handbook of Mathematical Functions, published by the National Bureau of Standards in 1964 and reprinted by Dover in 1965. Both editions continue to sell briskly and are widely cited in the scientific literature. However, with the many advances in the theory, computation and application of special functions that have occurred since 1960, a new standard reference is badly needed. NIST intends to satisfy this need by providing a Digital Library of Mathematical Functions (DLMF) as a free Web site together with an associated book and CDROM. The Web site will provide many capabilities that are impossible to provide in print media alone. 1
Symbolic evaluation of coefficients in Airytype asymptotic expansions
"... Computer algebra algorithms are developed for evaluating the coefficients in Airytype asymp totic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber par ..."
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Computer algebra algorithms are developed for evaluating the coefficients in Airytype asymp totic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function.
Uniform Asymptotics for the Incomplete Gamma Functions Starting From Negative Values of the Parameters
"... We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the i ..."
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We consider the asymptotic behavior of the incomplete gamma functions fl(\Gammaa; \Gammaz) and \Gamma(\Gammaa; \Gammaz) as a !1. Uniform expansions are needed to describe the transition area z a, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for fl(a; z) and \Gamma(a; z). The result is compared with Olver's uniform expansion for the generalized exponential integral. A numerical verification of the expansion is given in a final section.
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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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.
Nonvanishing of weight k modular Lfunctions with large level
 J. Ramanujan Math. Soc
, 1999
"... We will establish lower bounds in terms of the level for the number of holomorphic cusp forms of weight k> 2 whose various Lfunctions do not vanish at the central critical point. This work generalizes the work of W. Duke [1] which was for the case of weight 2. 1 ..."
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We will establish lower bounds in terms of the level for the number of holomorphic cusp forms of weight k> 2 whose various Lfunctions do not vanish at the central critical point. This work generalizes the work of W. Duke [1] which was for the case of weight 2. 1
Asymptotics and Numerics of Zeros of Polynomials That Are Related to Daubechies Wavelets
, 1997
"... We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical value ..."
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We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision. 1991 Mathematics Subject Classification: 33B20, 41A60, 65D20. Keywords & Phrases: Daubechies wavelets, zeros of high degree polynomials, incomplete beta functions, uniform asymptotic expansion. 1. Daubechies wavelets The polynomial formed by the first N terms of the binomial expansion (1 \Gamma y) \GammaN = 1 X k=0 ` k +N \Gamma 1 k ' y k = 1 X k=0 (N) k k! y k that is, the polynomial PN (y) = N \Gamma1 X k=0 ` k +N \Gamma 1 k ' y k = 1 +Ny + N(N + 1) 2 y 2 + : : : + ` 2N \Gamma 2 N \Gamma 1 ' y N \Gamma1 ; (1:1) where (a) k = \Gamma(a+k)=\Gamma(a), plays an important role in the construction of the compactly supp...
SOME PROPERTIES OF THE GAMMA AND PSI FUNCTIONS, WITH APPLICATIONS
"... Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some wellknown results for the volume Ωn of the unit ball B n ⊂ R n,thesu ..."
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Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some wellknown results for the volume Ωn of the unit ball B n ⊂ R n,thesurface area ωn−1 of the unit sphere S n−1, and some related constants. 1.