In (1.2.1) and (1.2.3) and are nonnegative integers and . In (1.2.2), (1.2.4), and (1.2.5) is a positive integer. See also §26.3(i).
1.2.1 | |||
For complex the binomial coefficient is defined via (1.2.6).
1.2.2 | |||
1.2.3 | |||
1.2.4 | |||
1.2.5 | |||
where is or according as is even or odd.
See also §26.3.
1.2.10 | |||
where = last term of the series = .
1.2.11 | |||
. | |||
Let be distinct constants, and be a polynomial of degree less than . Then
If are positive integers and , then there exist polynomials , , such that
1.2.16 | |||
To find the polynomials , , multiply both sides by the denominator of the left-hand side and equate coefficients. See Chrystal (1959a, pp. 151–159).
The arithmetic mean of numbers is
1.2.17 | |||
The geometric mean and harmonic mean of positive numbers are given by
1.2.18 | |||
1.2.19 | |||
If is a nonzero real number, then the weighted mean of nonnegative numbers , and positive numbers with
1.2.20 | |||
is defined by
1.2.21 | |||
with the exception
1.2.22 | |||
and . | |||
1.2.23 | ||||
1.2.24 | ||||
For , ,
1.2.25 | ||||
and
1.2.26 | |||
The last two equations require for all .