<p>The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by <i>p</i><i>n</i><sup>(λ)</sup>(<i>x</i>) instead of the standard notation <i>p</i><i>n</i><sup>(λ)</sup> (<i>x</i>/2, <i>π</i>/2), where λ > 0. The limiting case of these sequences of polynomials <i>p</i><i>n</i><sup>(0)</sup> (<i>x</i>) =lim<sub>λ→0</sub> <i>p</i><i>n</i><sup>(λ)</sup>(<i>x</i>), is obtained, and is shown to be an orthogonal sequence in the strip, <i>S</i> = {<i>z</i> ∈ ℂ : −1≤ℭ (<i>z</i>)≤1}.</p><p>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 <i>p</i><i>n</i><sup>(0)</sup> (<i>x</i>), 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 <i>p</i><i>n</i><sup>(0)</sup>(<i>x</i>) is obtained, and is found</p><p>to be ℙ = {{<i>p</i><i>n</i><sup>(λ)</sup> (<i>x</i>)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.</p><p>The polynomials {<i>p</i><i>n</i><sup>(λ)</sup> (<i>x</i>)}<sup>∞</sup><i>n</i><sub>=0</sub>, 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 <i>p</i><i>n</i><sup>(λ)</sup>(x) is orthogonal. </p><p>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.</p>
| Identifer | oai:union.ndltd.org:UPSALLA/oai:DiVA.org:uu-3501 |
| Date | January 2003 |
| Creators | Araaya, Tsehaye |
| Publisher | Uppsala University, Department of Mathematics, Uppsala : Matematiska institutionen |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, comprehensive summary, text |
| Relation | Uppsala Dissertations in Mathematics, 1401-2049 ; 27 |
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