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Bi-axial Gegenbauer functions of the first and second kind

Alan Common (1996)

Banach Center Publications

The classical orthogonal polynomials defined on intervals of the real line are related to many important branches of analysis and applied mathematics. Here a method is described to generalise this concept to polynomials defined on higher dimensional spaces using Bi-Axial Monogenic functions. The particular examples considered are Gegenbauer polynomials defined on the interval [-1,1] and the Gegenbauer functions of the second kind which are weighted Cauchy integral transforms over this interval of...

Linearization relations for the generalized Bedient polynomials of the first and second kinds via their integral representations

Shy-Der Lin, Shuoh-Jung Liu, Han-Chun Lu, Hari Mohan Srivastava (2013)

Czechoslovak Mathematical Journal

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials of the first and second kinds. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization...

Operators preserving orthogonality of polynomials

Francisco Marcellán, Franciszek Szafraniec (1996)

Studia Mathematica

Let S be a degree preserving linear operator of ℝ[X] into itself. The question is if, preserving orthogonality of some orthogonal polynomial sequences, S must necessarily be an operator of composition with some affine function of ℝ. In [2] this problem was considered for S mapping sequences of Laguerre polynomials onto sequences of orthogonal polynomials. Here we improve substantially the theorems of [2] as well as disprove the conjecture proposed there. We also consider the same questions for polynomials...

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