Gram Polynomials and the Kummer Function
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Journal of Approximation Theory 94 (1): 128 - 143 (1998)

Let φknk=0,n<m, be a family of polynomials orthogonal with respect to the positive semi-definite bilinear form(g, h)d:=1m∑j=1mg(xj)h(xj),xj:=−1+(2j−1)/m.These polynomials are known as Gram polynomials. The present paper investigates the growth of |φk(x)| as a function ofkandmfor fixedx∈−1, 1. We show that whenn⩽2.5m1/2, the polynomials in the family φknk=0are of modest size on −1, 1, and they are therefore well suited for the approximation of functions on this interval. We also demonstrate that if the degreekis close tom, andm⩾10, thenφk(x) oscillates with large amplitude for values ofxnear the endpoints of −1, 1, and this behavior makesφkpoorly suited for the approximation of functions on −1, 1. We study the growth properties of |φk(x)| by 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-standing conjectured inequality for the confluent hypergeometric function1F1, also known as Kummer's function, i.e., that1F1((1−a)/2, 1, t2)⩽1F1(1/2, 1, t2) for alla⩾0. In this paper we completely resolve this conjecture by verifying a generalization of the conjectured inequality with sharp constants.
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