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Umbral calculus and the Symmetric Meixner-Pollaczek polynomials
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
Manuscript (Other academic)
URN: urn:nbn:se:uu:diva-90615OAI: oai:DiVA.org:uu-90615DiVA: diva2:163041
Available from: 2003-05-16 Created: 2003-05-16 Last updated: 2010-01-13Bibliographically approved
In thesis
1. The Symmetric Meixner-Pollaczek polynomials
Open this publication in new window or tab >>The Symmetric Meixner-Pollaczek polynomials
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.

From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is found

to be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.

The polynomials {pn(λ) (x)}n=0, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal.

Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.

Place, publisher, year, edition, pages
Uppsala: Matematiska institutionen, 2003. 13 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 27
Mathematical analysis, Meixner-Pollaczek polynomial, Orthogonal polynomial, Polynomial operator, Inner product, Umbral calculus, Sheffer polynomial, Convolution type polynomial, Connection and linearization problem, 33C45, 05A40, 33D45, Matematisk analys
National Category
Mathematical Analysis
Research subject
urn:nbn:se:uu:diva-3501 (URN)91-506-1681-1 (ISBN)
Public defence
2003-06-06, Sal 247, MIC (Polacksbacken), Hus 2, Uppsala, 13:15
Available from: 2003-05-16 Created: 2003-05-16Bibliographically approved

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