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 .

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)?

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)

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).

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.

scholarly journals The expansion of functions in ultraspherical polynomials

1960 ◽  
Vol 1 (4) ◽  
pp. 428-438 ◽  
Author(s):  
David Elliott
Keyword(s):  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Ultraspherical Polynomial ◽  
Ultraspherical Polynomials ◽  
First Case ◽  
Image Position

The ultraspherical polynomial (x) of degree n and order λ is defined by for n = 0, 1, 2, …. Of these polynomials, the most commonly used are the Chebyshev polynomials Tn(x) of the first kind, corresponding to λ = 0; the Legendre polynomials Pn(x) for which λ = ½; and the Chebyshev polynomials Un(x) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since.

Polynomial Approximation and Growth of Generalized Axisymmetrig Potentials

1979 ◽  
Vol 31 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Peter A. McCoy
Keyword(s):  
Fourier Series ◽  
Initial Data ◽  
Polynomial Approximation ◽  
Jacobi Polynomials ◽  
Singular Line ◽  
Regular Solutions ◽  
Potential Equation ◽  
Ultraspherical Polynomials ◽  
Complete Set ◽  
Image Position

Generalized axisymmetric potentials Fα (GASP) are regular solutions to the generalized axisymmetric potential equation(1.1)in some neighborhood Ω of the origin where they are subject to the initial data(1.2)along the singular line y = 0. In Ω, these potentials may be uniquely expanded in terms of the complete set of normalized ultraspherical polynomials(1.3)defined from the symmetric Jacobi polynomials Pn(α, α)(ξ) of degree n with parameter α as Fourier series(1.4)

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.

Simple proof of a formula due to Racah

1963 ◽  
Vol 59 (2) ◽  
pp. 507-507 ◽  
Author(s):  
J. S. Griffith
Keyword(s):  
General Formula ◽  
Simple Proof ◽  
Legendre Polynomials ◽  
Closed Formula ◽  
Image Position ◽  
Special Case

After some manipulation from the general formula, Racah (1,2) obtained a closed formula for his V coefficient in the special case of all three mj- components zero. He also showed that the integral of the product of three Legendre polynomials followed at once from these special V coefficients. In this case the general formula ((2), eq. 10–14) becomeswhereand a, b, c are all non-negative integers. By changing the parameter of summation to (a+ b−c−p) it follows that is zero unless 2g = a+b+c is even (2).

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