q-Special Functions

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Proposal Number: DMS-0203282 PI: Dennis Stanton ABSTRACT Several topics in q-analysis will be studied. A family of generalizations of the Gaussian integral is considered which has several q's. These integrals are closely related to the Rogers-Ramanujan identities, and new results are found which are motivated by the integrals. An investigation of the root system version of the integrals will be undertaken. Some new classical q-orthogonal polynomials, q-Taylor series and combinatorial enumeration problems will also be considered. One of the simplest types of functions is a polynomial. More complicated functions such as sine, cosine, or exponentials can be approximated accurately using polynomials. The general theorem in calculus which allows such results is called Taylor's theorem, and it is used in applications throughout science. One of the objects of study in this proposal is a generalized approximation result, a q-Taylor theorem. It can be applied to functions currently under study in mathematics and statistical physics. Several related integrals, derivatives, and polynomials are also studied.