ABSTRACT Proposal: DMS- 9622816 PI: Askey The 3-j and 6-j symbols from angular momentum theory can be transformed into orthogonal polynomials. When this is done, new orthogonality conditions arise, with an absolutely continuous measure replacing the discrete one. The 9-j symbols are equivalent to orthogonal polynomials in two variables, but so far only the discrete orthogonality relation is known. It is time to look seriously at these functions, to try to find a double rather than a triple series representation for them, and then to find the missing absolutely continuous orthogonality relation. The corresponding work for q-9-j symbols will also be studied. In addition to this work and other work on one variable hypergeometric and basic hypergeometric series, some differential equations with 4 regular singular points will be studied. Lars Onsager's Ph.D. thesis from 1934 contained some interesting results for differential equations which are limits of these equations. This thesis will appear in his "Collected Papers" later this year. There are a number of interesting problems left open, including some on orthogonal polynomials that Askey will study. The interaction between mathematics and physics can lead to new insights in both areas. This has occurred in the quantum theory of angular momentum and in coding theory. In both fields, certain functions arose which turned out to be identical. These extended classical functions were studied extensively in the nineteenth and early twentieth centuries. The most general of these functions has recently arisen in the study of knots. The physics problems have been partly extended to several variables, but many open problems have not yet been solved. In the one variable setting, a very important aspect of these function was only found when some mathematicians studied them, while other ways of looking at them were discovered by physicists. Askey will be looking at these problems as a mathematician, and expects that this point of view wil l yield new information of use in both mathematics and physics.
