Positivity of basic sums of ultraspherical polynomials

Analysis ◽  
1998 ◽  
Vol 18 (4) ◽  
pp. 313-332 ◽  
Author(s):  
Gavin Brown ◽  
Stamatis Koumandos ◽  
Kun-Yang Wang
Keyword(s):  
Ultraspherical Polynomials

scholarly journals New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 74
Author(s):  
Waleed Mohamed Abd-Elhameed ◽  
Afnan Ali
Keyword(s):  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Algebraic Computation ◽  
Ultraspherical Polynomials ◽  
Current Article ◽  
Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

Some Characterizations of the Ultraspherical Polynomials

1968 ◽  
Vol 11 (3) ◽  
pp. 457-464 ◽  
Author(s):  
N.A. Al-Salam ◽  
W. A. Al-Salam
Keyword(s):  
Ultraspherical Polynomial ◽  
Ultraspherical Polynomials ◽  
Exact Degree ◽  
Image Position

Let be the nth ultraspherical polynomial. Also let . The following generating relation is well known (3, p.98).It can also be written as1.1This suggests the consideration of the class of polynomial sets {Qn(x), n = 0, 1, 2,…}, Qn(x) is of exact degree n and1.2

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 .

Convexity of the largest zero of the ultraspherical polynomials

1996 ◽  
Vol 4 (3) ◽  
pp. 275-278 ◽  
Author(s):  
C.G. Kokologiannaki ◽  
P.D. Siafarikas
Keyword(s):  
Ultraspherical Polynomials

The arithmetic of certain semigroups of positive operators

1968 ◽  
Vol 64 (1) ◽  
pp. 161-166 ◽  
Author(s):  
John Lamperti
Keyword(s):  
Finite Number ◽  
Positive Definite ◽  
Positive Operators ◽  
Linear Operators ◽  
Real Numbers ◽  
Ultraspherical Polynomials ◽  
Real Polynomials ◽  
Fixed Parameter ◽  
Summation Sign ◽  
Image Position

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measureand which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided thatHere the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that

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