Asymptotic Monotonicity of the Relative Extrema of Jacobi Polynomials

1994 ◽  
Vol 46 (06) ◽  
pp. 1318-1337 ◽  
Author(s):  
R. Wong ◽  
J.-M. Zhang

Abstract If μk,n (α,β) denotes the relative extrema of the Jacobi polynomial P(α,β) n(x), ordered so that μ k+1,n (α,β) lies to the left of μ k,n (α,β), then R. A. Askey has conjectured twenty years ago that for for k = 1,…, n — 1 and n = 1,2,=. In this paper, we give an asymptotic expansion for μ k,n (α,β) when k is fixed and n → ∞, which corrects an earlier result of R. Cooper (1950). Furthermore, we show that Askey's conjecture is true at least in the asymptotic sense.

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman

The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .


1970 ◽  
Vol 22 (3) ◽  
pp. 582-593 ◽  
Author(s):  
George Gasper

Let [3, p. 170, (16)](1.1)denote the Jacobi polynomial of order (α, β), α, β > – 1, and let g(k, m, n; α, β) be denned by(1.2)where Rn(α, β)(x) = Pn(α, β)(x)/Pn(α, β)(1). It is well known [1; 2; 4; 5; 6] that the harmonic analysis of Jacobi polynomials depends, at crucial points, on the answers to the following two questions.Question 1. For which (α, β) do we have(1.3)Question 2. For which (α, β) do we have(1.4)where G depends only on (α, β)?


1968 ◽  
Vol 16 (2) ◽  
pp. 101-108 ◽  
Author(s):  
J. S. Lowndes

Equations which may be regarded as extensions of the dual series equations discussed by Noble (1) and the present author (2) are the triple series equations of the first kindand the triple series equations of the second kindwhere f, f1, g, g1h and h1 are all known functions,is the Jacobi polynomial (3).


1969 ◽  
Vol 66 (1) ◽  
pp. 105-107 ◽  
Author(s):  
H. L. Manocha

In paper(1) it has been proved thatwhere the Jacobi polynomial is denned as ((3), p. 255)


1985 ◽  
Vol 37 (5) ◽  
pp. 979-1007 ◽  
Author(s):  
C. L. Frenzen ◽  
R. Wong

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:(1.1)where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).


Author(s):  
Richard Askey ◽  
George Gasper

The answers to many important questions in the harmonic analysis of orthogonal polynomials are known to depend on the determination of when formulas of the typesand their dualshold, where pn(x) and qn(x) are suitably normalized orthogonal polynomials or orthogonal polynomials multiplied by certain functions; e.g. e−pxLn(x).


1968 ◽  
Vol 64 (3) ◽  
pp. 687-690 ◽  
Author(s):  
H. L. Manocha

The writer in his paper (4) has shown thatwhere the Jacobi polynomial is defined as ((5), p. 255).


1968 ◽  
Vol 64 (3) ◽  
pp. 695-698 ◽  
Author(s):  
Paul R. Milch

Let be the Jacobi polynomial as defined by Szegö in (7) (see equation (4) below.) Carlitz in (2) presented among others the following formulaAlthough, as Carlitz claims, this formula may be derived directly from the definition of Jacobi polynomials, a probabilistic proof such as presented below may shed some new light on formula (1), as well as suggest probabilistic proofs for other similar formulas of Jacobi polynomials, e.g. those given by Manocha and Sharma in (4) and (5) and by Manocha in (3). In addition, it is quite possible that this method of proof will result in the derivation of some new formulas for Jacobi polynomials.


1985 ◽  
Vol 37 (3) ◽  
pp. 551-576 ◽  
Author(s):  
Mizan Rahman

The main objective of this paper is to find useful q-extensions of Feldheim's [6] bilinear formula for Jacobi polynomials, namely,1.1where the Appel function F4 is defined by1.2α1, α2, ρ are arbitrary complex parameters such that the series on both sides of (1.1) are convergent, and1.3is the Jacobi polynomial of degree k, (a)k being the usual shifted factorial.


1967 ◽  
Vol 63 (2) ◽  
pp. 457-459 ◽  
Author(s):  
H. L. Manocha

In a previous paper (3) the writer has provedwhere is a Jacobi polynomial defined as ((4) p.255)


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