On some formulas of W.Gosper and spectral properties of certain operators in weighted spaces

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.