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let and let

: and Then

:

and

:

In terms of the coefficients

:

Laguerre showed that the roots of this polynomial were bounded by

:

where

:

Inspection shows that is the mean of the roots and that b is the standard deviation of the roots.

Laguerre failed to notice this relationship with the means and standard deviations of the roots, being more interested in the bounds themselves. This relationship permits

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