Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients

1971 ◽  
Vol 70 (2) ◽  
pp. 243-255 ◽  
Author(s):  
Richard Askey ◽  
George Gasper
Keyword(s):  
Orthogonal Polynomials ◽  
Harmonic Analysis ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position ◽  
Polynomial Expansions

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).

Linearization of the Product of Jacobi Polynomials. II

1970 ◽  
Vol 22 (3) ◽  
pp. 582-593 ◽  
Author(s):  
George Gasper
Keyword(s):  
Harmonic Analysis ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position

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)?

Orthogonal Polynomials with Symmetry of Order Three

1984 ◽  
Vol 36 (4) ◽  
pp. 685-717 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Sphere ◽  
Jacobi Polynomials ◽  
Orthogonal Basis ◽  
Lower Degree ◽  
Surface Measure ◽  
Rotation Invariant ◽  
General Measure ◽  
Invariant Surface ◽  
Image Position

The measure (x1x2x3)2adm(x) on the unit sphere in R3 is invariant under sign-changes and permutations of the coordinates; here dm denotes the rotation-invariant surface measure. The more general measurecorresponds to the measureon the triangle(where ). Appell ([1] Chap. VI) constructed a basis of polynomials of degree n in v1, v2 orthogonal to all polynomials of lower degree, and a biorthogonal set for the case γ = 0. Later Fackerell and Littler [6] found a biorthogonal set for Appell's polynomials for γ ≠ 0. Meanwhile Pronol [10] had constructed an orthogonal basis in terms of Jacobi polynomials.

A Relation Between Ultraspherical and Jacobi Polynomial Sets

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman
Keyword(s):  
Jacobi Polynomial ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Ultraspherical Polynomials ◽  
Image Position ◽  
Special Case

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 .

scholarly journals Some triple series equations involving Jacobi polynomials

1968 ◽  
Vol 16 (2) ◽  
pp. 101-108 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Jacobi Polynomial ◽  
Present Author ◽  
Jacobi Polynomials ◽  
Dual Series Equations ◽  
Image Position

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).

Bilateral generating functions involving Jacobi polynomials

1969 ◽  
Vol 66 (1) ◽  
pp. 105-107 ◽  
Author(s):  
H. L. Manocha
Keyword(s):  
Generating Functions ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position ◽  
Bilateral Generating Functions

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

A Non-Negative Representation of the Linearization Coefficients of the Product of Jacobi Polynomials

1981 ◽  
Vol 33 (4) ◽  
pp. 915-928 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Hypergeometric Function ◽  
Jacobi Polynomials ◽  
Defining Relation ◽  
Linearization Problem ◽  
Image Position ◽  
Linearization Coefficients ◽  
Negative Representation

The problem of linearizing products of orthogonal polynomials, in general, and of ultraspherical and Jacobi polynomials, in particular, has been studied by several authors in recent years [1, 2, 9, 10, 13-16]. Standard defining relation [7, 18] for the Jacobi polynomials is given in terms of an ordinary hypergeometric function:with Re α > –1, Re β > –1, –1 ≦ x ≦ 1. However, for linearization problems the polynomials Rn(α,β)(x), normalized to unity at x = 1, are more convenient to use:(1.1)Roughly speaking, the linearization problem consists of finding the coefficients g(k, m, n; α,β) in the expansion(1.2)

Bilinear and trilinear generating functions for Jacobi polynomials

1968 ◽  
Vol 64 (3) ◽  
pp. 687-690 ◽  
Author(s):  
H. L. Manocha
Keyword(s):  
Generating Functions ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position

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

A probabilistic proof of a formula for Jacobi polynomials by L. Carlitz

1968 ◽  
Vol 64 (3) ◽  
pp. 695-698 ◽  
Author(s):  
Paul R. Milch
Keyword(s):  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Probabilistic Proof ◽  
Definition Of ◽  
Image Position

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.

A q-Extension of Feldheim's Bilinear Sum for Jacobi Polynomials and Some Applications

1985 ◽  
Vol 37 (3) ◽  
pp. 551-576 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Arbitrary Complex ◽  
Image Position

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.

Asymptotic Monotonicity of the Relative Extrema of Jacobi Polynomials

1994 ◽  
Vol 46 (06) ◽  
pp. 1318-1337 ◽  
Author(s):  
R. Wong ◽  
J.-M. Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position ◽  
Asymptotic Monotonicity

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.

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