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Gram Polynomials and the Kummer Function

1998; Elsevier BV; Volume: 94; Issue: 1 Linguagem: Inglês

10.1006/jath.1998.3181

1096-0430

Roger Barnard, Germund Dahlquist, Kent Pearce, Lothar Reichel, Kendall C. Richards,

Fractional Differential Equations Solutions

Let {φk}nk=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 {φk}nk=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.