GRAM POLYNOMIALS AND THE KUMMER FUNCTION

Let {phi(k)}(k=0)(n), n<m, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form (g, h)(d) := 1/m S igma(j=1)(m) g(x(j))h(x(j)), x(j) ;= -1 + (2j - 1)/m. These polynomial s are known as Gram polynomials. The present paper investigates the gr owth of \phi(k)(x)\ as a function of k and m for fixed x is an element of [ -1, 1]. We show that when n less than or equal to 2.5m(1/2), the polynomials in the family {phi(k)}(k=0)(n) are of modest size on [ -1 , 1], and they are therefore well suited for the approximation of func tions on this interval. We also demonstrate that if the degree k is cl ose to m, and m greater than or equal to 10, then phi(k)(x) oscillates with large amplitude for values of x near the endpoints of [ -1, 1], and this behavior makes phi(k) poorly suited for the approximation of functions on [ -1, 1]. We study the growth properties of \phi(k)(x)\ b y deriving a second order differential equation, one solution of which exposes the growth. The connection between Gram polynomials and this solution to the differential equation suggested what became a long-sta nding conjectured inequality for the confluent hypergeometric function F-1(1), also known as Kummer's function, i.e., that F-1(1)((1 - a)/2, 1, t(2)) less than or equal to F-1(1)(1/2, 1, t(2)) for all a greater than or equal to 0. In this paper we completely resolve this conjectu re by verifying a generalization of the conjectured inequality with sh arp constants. (C) 1998 Academic Press.