Mean iterations derived from transformation formulas for the hypergeometric function

Abstract

From Goursat"s transformation formulas for the hypergeometric function $F(\alpha,\beta,\gamma;z)$, we derive several double sequences given by mean iterations and express their common limits by the hypergeometric function. Our results are analogies of the fact that the arithmetic-geometric mean of 1 and $x\in (0,1)$ can be expressed as the reciprocal of $F \big( {1\over2},{1\over2},1;1-x^2 \big)$.