A Note on the Hypergeometric Mean Value
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A Note on the Hypergeometric Mean Value

DOI: 10.1007/BF03320978

Richards, Kendall C.

20180227

20180227

2001

Barnard, R. W., & Richards, K. C. (September 01, 2001). A Note on the Hypergeometric Mean Value. Computational Methods and Function Theory, 1, 1, 8188.

https://collections.southwestern.edu/s/suscholar/item/262

Recent efforts to obtain bounds for the complete elliptic integral 2 ·2F1−1 2, 1 2 ; 1; r2 in terms of power means and other related means have precipitated the search for similar bounds for the more general 2F1(α, β; γ; r). In an early paper, B. C. Carls considered the approximation of the hypergeometric mean values
( 2F1(−a, b; b + c; r))1/a in terms of means of order t, given by Mt(s, r) :=
{(1 − s) + s(1 − r)t}1/t. In this note, a refinement of one such approximation is established by first proving a general positivity result involving 3F2.

English

Computational Methods and Function Theory

Hypergeometric function

Generalized hypergeometric function

Means of order t

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