The discussion focuses on calculating binomial coefficients for negative integer exponents using Pascal's method. Specifically, it addresses the challenge of undefined factorials for negative integers when determining coefficients for expressions like (a + b) ^ -2. The solution involves utilizing the binomial series expansion, which provides a valid approach for negative exponents, as demonstrated by the formulas for (1 + x)^{r} and (1 + x)^{-r}. This method allows for the calculation of coefficients without encountering undefined factorials.
PREREQUISITESMathematicians, students studying algebra and combinatorics, and anyone interested in advanced applications of binomial coefficients and series expansions.
tommymato said:I'm using Pascal's (n choose k) method for calculating the coefficients of the terms of a binomial expansion. However, if the exponent is a negative integer, how can one use this method, seeing as factorials for negative integers are undefined.
For example, how could one determine the coefficients of (a + b) ^ -2
chisigma said:In the general case the binomial series is that which has here...
Binomial Series -- from Wolfram MathWorld
... in detail...
$\displaystyle (1 + x)^{r} = 1 + r\ x + \frac{r}{2}\ r\ (r-1) \ x^{2} + \frac{r}{6}\ r\ (r-1)\ (r-2)\ x^{3} + ... \ (1)$
$\displaystyle (1 + x)^{- r} = 1 + r\ x + \frac{r}{2}\ r\ (r+1) \ x^{2} + \frac{r}{6}\ r\ (r+1)\ (r+2)\ x^{3} + ... \ (2)$
Kind regards
$\chi$ $\sigma$