# Total Positivity of Mean Values and Hypergeometric Functions

@article{Carlson1983TotalPO,
title={Total Positivity of Mean Values and Hypergeometric Functions},
author={Bille C. Carlson and John L. Gustafson},
journal={Siam Journal on Mathematical Analysis},
year={1983},
volume={14},
pages={389-395}
}
• Published 1 March 1983
• Mathematics
• Siam Journal on Mathematical Analysis
The weighted power mean of two positive variables is strictly totally positive (STP) if its order t satisfies $- \infty 0$, which is equivalent to ${}_2 F_1$ with argument ${{1 - x} / y}$, is STP if $\alpha ,\beta \beta '$, and $\beta + \beta ' - \alpha$ are positive. With weaker restrictions this function is represented in a new way as a convolution. Higher order positivity is discussed for some other hypergeometric functions, including incomplete elliptic integrals.
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