Research InterestsMy overall area of interest is the theory of numbers. My research interests cover a wide range of different topics within this field including continued fractions, the theory of partitions, basic hypergeometric functions and arithmetical functions.
One part of number theory about which I have a keen interest is continued fractions. I am interested in both the arithmetical as well as the analytic and convergence theory of continued fractions. On the arithmetical side my work has resulted in the creation of a new algorithm related to finding rational approximations to real numbers. It also enhances the conceptual framework for understanding ``best'' rational approximations to real numbers.
I currently hold an NSF grant in the area of computational mathematics. My research here is really related to generalzations of the continued fraction algorithm and its associated best approximation properties. Another part of this research project is into the combinatorial aspects of continued fractions and the connections between the combinatorial properties of continued fractions and their approximation properties.
On the analytical and analytic number theory side of continued fractions, I have made a detailed study of extensions of the Rogers-Ramanujan (R-R) continued fraction. Closely related are the R-R identities which are of great number theoretic interest and they have been the stimulus of a considerable amount of research in recent years. The famous British mathematician G.H. Hardy described these formulas as ones that ``defeated him completely.'' In joint work with G.E. Andrews of Penn State, I extend the R-R continued fraction to six parameters in a way which meshes with G.N. Watson's generalization of the Rogers-Ramanujan identities.
On the convergence theory side of continued fractions I have made a study of modifications of convergence applied to q-continued fractions. This work explains some of the results found recently by K. Alladi. It shows that different limiting values can be achieved for a continued fraction, depending on how the convergence is defined. In fact I show that the values obtained by Alladi are natural when considered in the context of the vector space of solutions of the difference equation which the continued fraction defines. In general I suggest a new natural definition of convergence for q-continued fractions which gives continued fractions more versatility to express functions as limits of infinite processes.
Another large research project I am working on is the area of transformations of basic hypergeometric functions. Gauss introduced the hypergeometric function, which is now ubiquitous in mathematics and physics. Heine subsequently generalized Gauss's function to produce its q-analogue. Basic hypergeometric functions, or q-analogues, have recently become a focus of intense interest, as they are related to quantum groups, vertex operator algebras and lie algebras to name a few areas. They are also of central importance to additive number theory and modular forms. Heine found a transformation for his basic hypergeometric function which compressed a significant amount of information about the Gauss function. Later L.J. Rogers explained the Heine transformation by expanding Heine's function in a symmetrical way. Since Rogers' time few people have looked for such symmetric expansions. Extending Rogers' work I have found a large number of new symmetric expansions, each of which contains a wealth of transformations for special functions. This work is intimately tied with orthogonal and biorthogonal polynomials, a connection I have been exploiting which has led to new results such as general Askey-Wilson type integrals.
Another area in which I work is combinatorial number theory, especially the theory of partitions. Here I discovered a generalization of partitions of positive integers, namely partitions with restricted gap conditions where the gaps are partially filled with certain integers. I am exploring bijections between classes of these and classes of ordinary partitions. Among other results, I have found bijections between my new partitions and several previously studied types of partitions including a bijection with ordinary partitions as well as one with partitions into odd parts. I am considering extending some of these results to multidimensional continued fractions.
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