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Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions
 MATHEMATICS OF COMPUTATION
, 2007
"... Three term recurrence relations yn+1 +bnyn +anyn−1 =0 can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for do ..."
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Three term recurrence relations yn+1 +bnyn +anyn−1 =0 can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their role for finite values of n with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal. This phenomenon, related to Gautschi’s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions, is shown to be a general situation which takes place for recurrences with an negative and bn changing sign once. We analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations (modified Bessel functions, confluent and Gauss hypergeometric functions) and discuss the implication of such transitory behaviour on the numerical stability of recursion.
Basic methods for computing special functions
, 2009
"... This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequentl ..."
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This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.
THIRD ORDER NEWTON’S METHOD FOR ZERNIKE POLYNOMIAL ZEROS
, 705
"... Abstract. The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of circular pupils. To calculate the roots of Zernike polynomials, we optimize the generic iterativ ..."
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Abstract. The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of circular pupils. To calculate the roots of Zernike polynomials, we optimize the generic iterative numerical Newton’s Method that iterates on zeros of functions with third order convergence. The technique is based on rewriting the polynomials as Gauss hypergeometric functions, reduction of second order derivatives to first order derivatives, and evaluation of some ratios of derivatives by terminating continued fractions. A PARI program and a short table of zeros complete up polynomials of 16th order are included. 1. Classical Orthogonal Polynomials: Hofsommer’s Newton Method The generic third order Newton’s Method to compute roots f(x) = 0 numerically improves solutions xi → xi+1 = xi + ∆x iteratively, starting from initial guesses, via computation of corrections (1) ∆x = − f(x) f ′ (x) 1 + f(x)
A ̀ LA CARTE RECURRENCE RELATIONS FOR CONTINUOUS AND DISCRETE HYPERGEOMETRIC FUNCTIONS
"... Abstract. We show how, using the constructive approach for special functions introduced by Nikiforov and Uvarov, one can obtain recurrence relations for the hypergeometrictype functions not only for the continuous case but also for the discrete and qlinear cases, respectively. Some applications i ..."
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Abstract. We show how, using the constructive approach for special functions introduced by Nikiforov and Uvarov, one can obtain recurrence relations for the hypergeometrictype functions not only for the continuous case but also for the discrete and qlinear cases, respectively. Some applications in Quantum Physics are discussed.
NUMERICAL METHODS FOR THE COMPUTATION OF THE CONFLUENT AND GAUSS HYPERGEOMETRIC FUNCTIONS∗
"... Abstract. The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable re ..."
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Abstract. The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, GaussJacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide ‘roadmaps ’ with our recommendation for which methods should be used in each situation. Key words. Computation of special functions, confluent hypergeometric function, Gauss hypergeometric function AMS subject classifications. Primary: 33C05, 33C15; Secondary: 41A58, 41A60 1. Introduction. The
STRUCTURAL AND RECURRENCE RELATIONS FOR HYPERGEOMETRICTYPE FUNCTIONS BY NIKIFOROVUVAROV METHOD∗
"... Abstract. The functions of hypergeometrictype are the solutions y = yν(z) of the differential equation σ(z)y′ ′ + τ(z)y ′ + λy = 0, where σ and τ are polynomials of degrees not higher than 2 and 1, respectively, and λ is a constant. Here we consider a class of functions of hypergeometric type: thos ..."
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Abstract. The functions of hypergeometrictype are the solutions y = yν(z) of the differential equation σ(z)y′ ′ + τ(z)y ′ + λy = 0, where σ and τ are polynomials of degrees not higher than 2 and 1, respectively, and λ is a constant. Here we consider a class of functions of hypergeometric type: those that satisfy the condition λ+ ντ ′ + 1 2 ν(ν − 1)σ′ ′ = 0, where ν is an arbitrary complex (fixed) number. We also assume that the coefficients of the polynomials σ and τ do not depend on ν. To this class of functions belong Gauss, Kummer, and Hermite functions, and also the classical orthogonal polynomials. In this work, using the constructive approach introduced by Nikiforov and Uvarov, several structural properties of the hypergeometrictype functions y = yν(z) are obtained. Applications to hypergeometric functions and classical orthogonal polynomials are also given. Key words. hypergeometrictype functions, recurrence relations, classical orthogonal polynomials AMS subject classifications. 33C45, 33C05, 33C15 1. Introduction. When