The Symmetric Meixner-Pollaczek polynomials

Sammanfattning: The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(?)(x) instead of the standard notation pn(?) (x/2, ?/2), where ? > 0. The limiting case of these sequences of polynomials pn(0) (x) =lim??0 pn(?)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ? ? : ?1?? (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 foundto be ? = {{pn(?) (x)} =0 : ? ? R}. The major properties of these sequences of polynomials are studied.The polynomials {pn(?) (x)}?n=0, ? < 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 ? ? 0, an associated nonstandard inner product is defined with respect to which pn(?)(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.