Hokkaido Mathematical Journal

HATTORI Ryohei, KATO Takayuki, MATSUMOTO Keiji,

Mean iterations derived from transformation formulas for the hypergeometric function.

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
MSC(Secondary)26A18
Uncontrolled Keywordshypergeometric function, mean iteration.