where is or according as is even or odd.
See also §26.3.
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 .