where = last term of the series = . Let be distinct constants,
and be a polynomial of degree less than . Then and is the -th
derivative of (§1.4(iii)). If are positive integers and , then
there exist polynomials , , such that To find the polynomials ,
, multiply both sides by the denominator of the left-hand side
and equate coefficients. See Chrystal (1959, pp. 151–159). The
arithmetic mean of numbers is The geometric mean and harmonic
mean of positive numbers are given by If is a nonzero real
number, then the weighted mean of nonnegative numbers , and
positive numbers with is defined by with the exception For , ,
The last two equations require for all .