Binomial series
Negative binomial series

Closely related is the negative binomial series defined by the MacLaurin series for the function g (x) = (1 − x) − α {\displaystyle g(x)=(1-x)^{-\alpha }}, where α ∈ C {\displaystyle \alpha \in \mathbb {C} } and | x | < 1 {\displaystyle |x|<1}. Explicitly,

which is written in terms of the multiset coefficient

When α is a positive integer, several common sequences are apparent. The case α = 1 gives the series 1 + x + x + x +..., where the coefficient of each term of the series is simply 1. The case α = 2 gives the series 1 + 2 x + 3 x + 4 x +..., which has the counting numbers as coefficients. The case α = 3 gives the series 1 + 3 x + 6 x + 10 x +..., which has the triangle numbers as coefficients. The case α = 4 gives the series 1 + 4 x + 10 x + 20 x +..., which has the tetrahedral numbers as coefficients, and similarly for higher integer values of α.

The negative binomial series includes the case of the geometric series, the power series 1 1 − x = ∑ n = 0 ∞ x n {\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}} (which is the negative binomial series when α = 1 {\displaystyle \alpha =1}, convergent in the disc | x | < 1 {\displaystyle |x|<1}) and, more generally, series obtained by differentiation of the geometric power series: 1 (1 − x) n = 1 (n − 1) ! d n − 1 d x n − 1 1 1 − x {\displaystyle {\frac {1}{(1-x)^{n}}}={\frac {1}{(n-1)!}}{\frac {d^{n-1}}{dx^{n-1}}}{\frac {1}{1-x}}} with α = n {\displaystyle \alpha =n}, a positive integer.
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