No abstract

We extend the Stern sequence, sometimes also called Stern's diatomic sequence, to polynomials with coefficients 0 and 1 and derive various properties, including a generating function. A simple iteration for quotients of consecutive terms of the Stern sequence, recently obtained by Moshe Newman, is extended to this polynomial sequence. Finally we establish connections with Stirling numbers and Chebyshev polynomials, extending some results of Carlitz. In the process we also obtain some new results and new proofs for the classical Stern sequence.

Let K be a compact set, % a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and / a continuous real-valued function on Kx K. We study the problem of determining for which /x e% (if any) the energy integral 1(K, fi) = Siefi(ßx' y)df4.x)df/.y) is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere. In particular, when % is atomic we obtain good upper bounds for the sums of powers of all (?) distances determined by n points on the surface of a sphere. We make use of results from Schoenberg's theory of metric embedding, and of techniques devised by Polya and Szegö for the calculation of transfinite diameters.

Abstract.An upper bound for the sum of the Ath powers of all distances determined by N points on a unit sphere is given for

An upper bound for the sum of the Ath powers of all distances determined by N points on a unit sphere is given for Let plt ■ • •, pN be points on the unit sphere UmofE'", the w-dimensional Euclidean space. Let