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It is to be understood here that du(x) has moments of all orders on [0, °°), and that n < N 1 if du(x) is a discrete distribution over only N points. Using different arguments, Askey [1] and the author [4] have shown that the conclusion of this theorem follows from classical results if c is a positive integer, and, allowing for a difference in normalization, results obtained in [4] imply also that the coefficients in the "inverse" expansion, pn(x) = 2"_ 0 bjnqj(x)9 satisfy ( \) ^b-n > 0 if c is a positive integer, but that this does not hold for all positive c. It suffices to prove Theorem 1 for 0 < c < 1, since its conclusion follows for any positive c from finitely many successive applications of this restricted result. We confine our attention to this case. The roots of pt(x) and qt(x) are positive; therefore, (1) and Descartes' rule of signs imply that

Proof of a conjecture of Askey on orthogonal expansions with positive coefficients