# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 38 (2009) pp.563-586

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### 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 $F \big( {1\over2},{1\over2},1;1-x^2 \big)$.

MSC(Primary) 33A25 26A18 hypergeometric function, mean iteration.