Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

Cohl, Howard S.

doi:10.3842/sigma.2013.042

Cited by 15 publications

(20 citation statements)

References 38 publications

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“…Formula (8.9) can also be obtained as the special case ν = −n of formula (3.1) in Cohl [5]. In that formula he gives an explicit expansion of (z − x) −ν in terms of Jacobi polynomials P (α,β) n (x), where the expansion coefficients turn out to be constant multiples of the expressions (z − 1) α+1−ν (z + 1) β+1−ν Q (α+1−ν,β+1−ν) n+ν−1 (z) (the Q-functions being Jacobi functions of the second kind).…”

mentioning

confidence: 99%

Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials

Cagliero

Koornwinder

2015

Journal of Approximation Theory

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For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Román. The twoparameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomials, unrelated to the family just mentioned, was given by J. Koekoek & R. Koekoek (1999). We show that this last family is a limit case of a pair of connection relations between Askey-Wilson polynomials having one of their four parameters in common.

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confidence: 99%

Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials

Cagliero

Koornwinder

2015

Journal of Approximation Theory

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“…For fixed z, this series converges uniformly in x ∈ [−1, 1] (see, e.g., [26, Chapter IX, Theorem 9.1.1]) and as shown in [5], it is valid for a set of parameters λ, γ, β containing R * + ×(−1, ∞) 2 . However, we shall restrict ourselves to the latter which is sufficient for our purposes and subsequent computations.…”

Section: Generalized Stieltjes Transform Of Beta Distributionsmentioning

confidence: 99%

“…Recently, this task was achieved in [5] for the family of Jacobi polynomials whence the generalized Stieltjes transform of any beta distribution in [−1, 1] follows after a simple integration. Using linear and quadratic transformations of the Gauss hypergeometric function, we retrieve the few examples of (1.1) given in [6] and prove for two larger classes of beta distributions that their generalized Stieltjes transforms are elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution.…”

Section: Remindermentioning

confidence: 99%

Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples

Демни

2016

SIGMA

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Abstract. For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. As to the free Poisson distribution, it corresponds to the product of two independent Beta distributions in [0, 1] while another example of Beta distributions in [−1, 1] is found and is related with the Shrinkage process. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distribution.

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“…We have since demonstrated that this technique is valid for a larger class of hypergeometric orthogonal polynomials. For instance, in [2], we applied this same technique to the Jacobi polynomials and in [5], we extended this technique to many generating functions for the Jacobi, Gegenbauer, Laguerre, and Wilson polynomials.…”

Section: Introductionmentioning

confidence: 99%

On a generalization of the Rogers generating function

Cohl

Costas-Santos

Wakhare

2019

Journal of Mathematical Analysis and Applications

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We derive a generalized Rogers generating function and corresponding definite integral, for the continuous q-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the Rogers generating function by Ismail & Simeonov expanded in terms of Askey-Wilson polynomials, we derive corresponding generalized expansions for the continuous q-Jacobi, and Wilson polynomials with two and four free parameters respectively. Comparing the coefficients of the Askey-Wilson expansion to our continuous q-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric series connecting 2 φ 1 and 8 φ 7 .

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Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

Cited by 15 publications

References 38 publications

Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials

Explicit matrix inverses for lower triangular matrices with entries involving Jacobi polynomials

Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples

On a generalization of the Rogers generating function

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