Multiple Wilson and Jacobi–Piñeiro polynomials

Beckermann, Bernhard; Coussement, J.; Assche, Walter Van

doi:10.1016/j.jat.2004.12.001

Cited by 33 publications

(37 citation statements)

References 14 publications

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“…A similar phenomenon occurs for ζ (3), where one recovers Apéry's celebrated approximations, also given in [1]. Furthermore, in [12], the second author adapted this method to produce a sequence of fast converging rational approximations u n /v n for Catalan's constant G = ∞ k=0 (−1) k /(2k + 1) 2 (the rôle of the Bernoulli's numbers being played by Euler's numbers); although the irrationality of G could not be deduced from these approximations, they were found to be the same as the approximationsû n /v n to G previously obtained in [13] by means of a completely different method, based on hypergeometric series (the proof of this coincidence is quite long and intricate).…”

Section: Introductionsupporting

confidence: 71%

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Remainder Pade Approximants for the Exponential Function

Prévost

Rivoal

2006

Constr Approx

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Abstract. Following earlier research of ours, we propose a new method for obtaining the complete Padé table of the exponential function. It is based on an explicit construction of certain Padé approximants not for the usual power series for exp at 0 but for a formal power series related in a simple way to the remainder term of the power series for exp. This surprising and non trivial coincidence is proved more generally for type II simultaneous Padé approximants for a family (exp(a j z)) j=1,...,r with distinct complex a's and we recover Hermite's classical formulae. The proof uses certain discrete multiple orthogonal polynomials recently introduced by Arvesú, Coussement and van Assche, which generalise the classical Charlier orthogonal polynomials.

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Section: Introductionsupporting

confidence: 71%

“…, p r ). Fortunately, an explicit form for these polynomials already exists in the literature, under the name of multiple Charlier polynomials (see [2,3] but with a different normalization):…”

Section: Proof Of Theoremmentioning

confidence: 99%

Remainder Pade Approximants for the Exponential Function

Prévost

Rivoal

2006

Constr Approx

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“…It is obvious that one has to worry about the interplay between indices and integration contours, a subject studied in [5,6]. There the authors claim dependence of the orthogonality properties on the indices and the integration contours.…”

Section: The Trigonometric Rosen-morse Potential As Complexifiedmentioning

confidence: 99%

The trigonometric Rosen–Morse potential in the supersymmetric quantum mechanics and its exact solutions

Compean

Kirchbach

2005

J. Phys. A: Math. Gen.

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The analytic solutions of the one-dimensional Schrödinger equation for the trigonometric RosenMorse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention to the fact that the complex Jacobi polynomials have non-trivial orthogonality properties which make them uncomfortable for physics applications.Instead we here solve above equation in terms of real orthogonal polynomials. The new solutions are used in the construction of the quantum-mechanical superpotential.

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“…Groenevelt [12] used Wilson functions, which is a family of transcendental solutions of the aforementioned second-order SturmLiouville Wilson difference equation linearly independent to Wilson polynomials, as the kernel of a new integral transform called the Wilson function transform. Wilson polynomials have also been extended to a multiple-parameter version [4] using essentially the same tool of Jacobi transform that was used in [12]. Wilson polynomials also have applications in various aspects from theoretical physics to birth and death processes, see for examples [5], [19], [21], [28] and [29].…”

Section: Introductionmentioning

confidence: 99%

Nevanlinna theory of the Wilson divided-difference operator

Cheng¹,

Chiang²

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. Sitting at the top level of the Askey-scheme, Wilson polynomials are regarded as the most general hypergeometric orthogonal polynomials. Instead of a differential equation, they satisfy a second order Sturm-Liouville type difference equation in terms of the Wilson divideddifference operator. This suggests that in order to better understand the distinctive properties of Wilson polynomials and related topics, one should use a function theory that is more natural with respect to the Wilson operator. Inspired by the recent work of Halburd and Korhonen, we establish a full-fledged Nevanlinna theory of the Wilson operator for meromorphic functions of finite order. In particular, we prove a Wilson analogue of the lemma on logarithmic derivatives, which helps us to derive Wilson operator versions of Nevanlinna's Second Fundamental Theorem, some defect relations and Picard's Theorem. These allow us to gain new insights on the distributions of zeros and poles of functions related to the Wilson operator, which is different from the classical viewpoint. We have also obtained a relevant five-value theorem and Clunie type theorem as applications of our theory, as well as a pointwise estimate of the logarithmic Wilson difference, which yields new estimates to the growth of meromorphic solutions to some Wilson difference equations and Wilson interpolation equations.

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Multiple Wilson and Jacobi–Piñeiro polynomials

Cited by 33 publications

References 14 publications

Remainder Pade Approximants for the Exponential Function

Remainder Pade Approximants for the Exponential Function

The trigonometric Rosen–Morse potential in the supersymmetric quantum mechanics and its exact solutions

Nevanlinna theory of the Wilson divided-difference operator

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