The many limits of mixed means

Abstract

The sequences introduced by Carlson (1971) are variants of the Gauss arithmetic geometric sequences (which have been elegantly discussed by D. A. Cox (1984, 1985)). Given (complex)a 0,b 0 we define $$a_{n + 1} : = \tfrac{1}{2}(a_n + b_n ),b_{n + 1} : = \sqrt {(a_n a_{n + 1} )} $$ and our program is to discuss the convergence and limits of the sequences {a n }, {b n } when the determinations of the square root are made according to an assigned pattern. The original assigned pattern was all positive. Carlson's original discussion made use of an invariant integral in the case whena 0,b 0 were non-negative and all determinations were positive. The discussion of the general Gaussian case used a parametrization of the sequences by thetaconstants. Our discussion of the Carlson case will use a parameterization of the sequences by lemniscatefunctions, although it could equally be written in terms of thetafunctions (in the lemniscate case). The complex multiplication of these functions is used essentially. We made considerable use of computers and we record some sample results.