TAOCP 7.2.1.5 Exercise 54
From equation (40) of the text, the parameters \(\alpha\) and \(\beta\) are given by \[ \alpha = s(\coth s + 1), \qquad \beta = s(\coth s - 1), \] where \(s = \sigma/2\).
Section 7.2.1.5: Generating All Set Partitions
Exercise 54. [HM20] If $\alpha$ and $\beta$ are defined by (40), show that their arithmetic and geometric means are respectively $\dfrac{\alpha+\beta}{2}=s,\coth s$ and $\sqrt{\alpha\beta}=s,\csch s$, where $s=\sigma/2$.
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Solution
From equation (40) of the text, the parameters (\alpha) and (\beta) are given by
[ \alpha = s(\coth s + 1), \qquad \beta = s(\coth s - 1), ]
where (s = \sigma/2). Their arithmetic mean is therefore
[ \frac{\alpha + \beta}{2} = \frac{s(\coth s + 1) + s(\coth s - 1)}{2} = s\coth s. ]
Their product is
[ \alpha\beta = s^2(\coth^2 s - 1) = s^2\csch^2 s, ]
so the geometric mean is
[ \sqrt{\alpha\beta} = s\csch s. ]
This completes the proof. ∎