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)

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).

Some bilinear generating functions for Jacobi polynomials

1967 ◽  
Vol 63 (2) ◽  
pp. 457-459 ◽  
Author(s):  
H. L. Manocha
Keyword(s):  
Generating Functions ◽  
Jacobi Polynomial ◽  
Jacobi Polynomials ◽  
Image Position

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

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).

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).

scholarly journals Some extensions of Bateman's product formulas for the Jacobi polynomials

1995 ◽  
Vol 8 (4) ◽  
pp. 423-428 ◽  
Author(s):  
Ming-Po Chen ◽  
H. M. Srivastava
Keyword(s):  
Generating Functions ◽  
Jacobi Polynomials ◽  
Bilateral Generating Functions ◽  
Product Formulas

The authors derive generalizations of some remarkable product formulas of Harry Bateman (1882-1946) for the classical Jacobi polynomials. They also show how the results considered here would lead to various families of linear, bilinear, and bilateral generating functions for the Jacobi and related polynomials.

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.

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