Special Functions and Orthogonal Polynomials

Beals, Richard; Wong, R.

doi:10.1017/cbo9781316227381

Cited by 85 publications

(94 citation statements)

References 218 publications

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“…This result has been proved in [19]. Because we need the information given in (16), to make the paper self-contained we give a brief derivation of the above proposition by following the analysis in [1]. All results in the subsequent sections are appearing for the first time.…”

Section: Zero Distributionmentioning

confidence: 88%

“…It is well‐known that some second‐order linear ordinary differential equations have polynomial solutions; for instance, those satisfied by the three classical orthogonal polynomials. The Pseudo‐Jacobi (P‐J) polynomials can be introduced as the eigenfunctions of the symmetric differential operator defined by

\begin{matrix} true \\ right L = p \frac{d^{2}}{d x^{2}} + \frac{{(p w)}^{'}}{w} \frac{d}{d x}, \end{matrix}

where

p (x)

is a polynomial of degree 2 and

w (x)

is a continuous function called the weight function associated with L ; see [, p. 56]. This operator can admit orthogonal polynomials as its eigenfunctions by proper choices of p and w , and the three classical ones (namely, Hermite, Laguerre, Jacobi) correspond to

p = 1, x, 1 - x^{2}

, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…This operator can admit orthogonal polynomials as its eigenfunctions by proper choices of p and w , and the three classical ones (namely, Hermite, Laguerre, Jacobi) correspond to

p = 1, x, 1 - x^{2}

, respectively. Another case is when

p (x)

has distinct complex roots; after normalization, we can take

p = x^{2} + 1

; see [, p. 58]. A necessary condition that the operator L takes the space of real quadratic polynomials into itself is that

{(p w)}^{'} / w

p w^{'} / w

is a polynomial of degree at most 1.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotics of Pseudo‐Jacobi Polynomials with Varying Parameters

Song

Wong

2017

Stud Appl Math

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In this paper, we study the asymptotic behavior of the Pseudo‐Jacobi polynomials Pnfalse(z;a,bfalse) as n→∞ for z in the whole complex plane. These polynomials are also known as the Romanovski–Routh polynomials. They occur in quantum mechanics, quark physics, and random matrix theory. When the parameter a is fixed or a>−n, there is no real‐line orthogonality. Here, we consider the case when the parameters a and b depend on n; more precisely, we assume a=−(An+A0),A>1 and b=Bn+B0, where A,B,A0,B0 are real constants. Our main tool is the asymptotic method developed for differential equations with a large parameter.

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Section: Zero Distributionmentioning

confidence: 88%

\begin{matrix} true \\ right L = p \frac{d^{2}}{d x^{2}} + \frac{{(p w)}^{'}}{w} \frac{d}{d x}, \end{matrix}

where

p (x)

is a polynomial of degree 2 and

w (x)

p = 1, x, 1 - x^{2}

, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…This operator can admit orthogonal polynomials as its eigenfunctions by proper choices of p and w , and the three classical ones (namely, Hermite, Laguerre, Jacobi) correspond to

p = 1, x, 1 - x^{2}

, respectively. Another case is when

p (x)

has distinct complex roots; after normalization, we can take

p = x^{2} + 1

; see [, p. 58]. A necessary condition that the operator L takes the space of real quadratic polynomials into itself is that

{(p w)}^{'} / w

p w^{'} / w

is a polynomial of degree at most 1.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotics of Pseudo‐Jacobi Polynomials with Varying Parameters

Song

Wong

2017

Stud Appl Math

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“…The interested reader may want to refer to [1][2][3] for full accounts of this fascinating area of orthogonal polynomials.…”

Section: (7)mentioning

confidence: 99%

Representing Sums of Finite Products of Chebyshev Polynomials of Third and Fourth Kinds by Chebyshev Polynomials

et al. 2018

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“…where H p (x) are the conventional Hermite polynomials. [27,28] We use now Eq. (11), and observing that ∂ 2n…”

Section: Cmentioning

confidence: 99%

Quasi‐Relativistic Heat Equation via Lévy Stable Distributions: Exact Solutions

et al. 2017

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We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one-sided Lévy stable distributions. We note a natural appearance of Bessel polynomials which allow one the obtention of closed form solutions for a number of initial conditions. The resulting relativistic diffusion is slower than the non-relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and is compared with the non-relativistic case. In this note we develop an alternative point of view which takes advantage of the concepts of anomalous diffusion [7] governed by the evolution equations involving square root pseudo-differential operators. We postulate and study in this note the following one-dimensional differential equation for the distribution function ψ(ξ, τ ) which we believe would be appropriate in the relativistic (R) situation: which is formally of infinite order in ∂ 2 x . For the purpose of this work we shall refer to Eq. (3) as a relativistic heat equation. In Quantum Mechanics the difficulty to treat the square root in Eqs.(1) and (3) (so-called pseudodifferential operators [11,12]) is usually resolved by introducing the wave functions with several components, like in Pauli and in Dirac equations [11,12]. However the interest for equations involving pseudo-differential operators arose even before the inception of the Dirac equation. It can be traced back to H. Weyl [13]. The fact that Eqs. (1) and (3) correctly reproduce the NR limit is reminiscent of the R extension of the analogy between the Schrödinger and the heat equations, carried out in 1984 in [4], by identifying the underlying Poisson proarXiv:1610.05605v1 [math-ph]

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Special Functions and Orthogonal Polynomials

Cited by 85 publications

References 218 publications

Asymptotics of Pseudo‐Jacobi Polynomials with Varying Parameters

Asymptotics of Pseudo‐Jacobi Polynomials with Varying Parameters

Representing Sums of Finite Products of Chebyshev Polynomials of Third and Fourth Kinds by Chebyshev Polynomials

Quasi‐Relativistic Heat Equation via Lévy Stable Distributions: Exact Solutions

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