<p>The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by <i>p</i><i>n</i>^(λ)(<i>x</i>) instead of the standard notation <i>p</i><i>n</i>^(λ) (<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>⁽⁰⁾ (<i>x</i>) =lim_λ→0 <i>p</i><i>n</i>^(λ)(<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>⁽⁰⁾ (<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>⁽⁰⁾(<i>x</i>) is obtained, and is found</p><p>to be ℙ = {{<i>p</i><i>n</i>^(λ) (<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>^(λ) (<i>x</i>)}^∞<i>n</i>₌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 <i>p</i><i>n</i>^(λ)(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>

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 &gt; 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 &lt; 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.