Gasper 9401452 The award supports mathematical research on fundamental properties of hypergeometric and q-(basic) hypergeometric series, classical and non-classical orthogonal polynomials, and on the development of new methods for proving that certain orthogonal polynomials and entire functions have only real roots. In particular, investigations will be made into the nonnegativity of the coefficients in linearization formulas, the positivity of kernels in product formulas (both of which lead to new convolution structures and hypergroups), and necessary and sufficient multiplier conditions (via norms involving fractional differences) for certain systems of orthogonal polynomials. Continuing work will also be done on sums and integrals of squares of real-valued functions and their application in deriving explicit identities which prove that certain entire special functions have only real roots, and in deriving new families of inequalities, which also prove that all of the rootsare real. The theory of special functions and orthogonal polynomials has fruitful connections with many branches of mathematical analysis and applied mathematics, such as differential equations, operator theory, numerical analysis, integration, continued fractions and control theory to name a few. This particular program makes interesting connections with some of the most outstanding mathematical questions considered in the mainstream of interest at this time. Of particular note is the study of roots of entire functions which relates directly to the Riemann hypthesis. ***
