Vol. 89, No. 1, 1980

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Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products

Kenneth Barry Stolarsky

Vol. 89 (1980), No. 1, 209–227

The natural logarithm of z can be written as an infinite product involving iterated square roots of z. A Vieta product is defined to be a more general infinite product involving z raised to arbitrary fractional powers. Restricted to the unit circle, Vieta products generalize infinite cosine products studied by Salem and others in connection with PV -numbers. Vieta products are shown to have conformal mapping, monotonicity, and growth properties very similar to those of the natural logarithm. By using certain properties of Eulerian polynomials, the exponents of z in a Vieta product are shown to be unique in a strong sense.

Mathematical Subject Classification 2000
Primary: 30E99
Secondary: 40A20
Received: 17 September 1979
Published: 1 July 1980
Kenneth Barry Stolarsky