Content area
Abstract
By using his path invariant method and the symbolic algebra package Macsyma, R. William Gosper discovered many interesting identities. In Chapter 1 we prove some of them in a more conventional way and use our approach to prove some of his conjectures. It turned out that what is behind some of the summations are one-step iterations and what is behind some of the continued fraction identities are 3 term recurrence relations.
It is known that most general classical orthogonal polynomials are Askey-Wilson polynomials. They satisfy a second order equation in ${\cal D}\sb{q}$, ${\cal D}\sb{q}$ being the Askey-Wilson operator. This naturally led to the question of investigating special properties of ${\cal D}\sb{q}$ in various weighted spaces. In Chapter 2, we consider the operator $D={d\over dx}$, on the ultraspherical space $L\sp2\lbrack (1-x\sp2)\sp{\nu-1/2}dx$) and the Jacobi space $L\sp2\lbrack (1-x\sp\alpha)(1+x)\sp\beta dx$). The point spectra are zeros of Bessel functions of the first kind and zeros of Coulomb function respectively. We find the point spectrum of ${\cal D}\sb{q}$ to be the set of zeros of Jackson q Bessel functions. We also have a new q-generalization of the exponential function and some new expansion in terms of q-ultraspherical polynomials.
