In this case, the binomial coefficient
for a negative value is computed as follows:
where [F.sub.m](n) = n!/[m!(n - m)!] is the binomial coefficient
and i = [square root of -1].
The extension of the q- binomial coefficient
to non-integer value n is allowed via the [[GAMMA].sub.q] function defined for 0 < q < 1 as
We see that the estimated negative binomial coefficient
on the cultural distance variable is -0.3269.
Note the similarity between this and the expression in (1.1): the power [q.sup.[epsilon]] in (1.1) is replaced here by the central binomial coefficient
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The corresponding quotient [p.sub.n](I) is called Pascal descent polynomial and is in many aspects a generalization of the notion of the signed binomial coefficient
. We study the polynomials [p.sub.n](1) in Section 5 and give a recursion formula for their calculation.
By this, e.g., one immediately sees that each binomial coefficient
of the form ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is even, and if a and are even, then
Proposition 1: Every triangular number is a binomial coefficient
(19) The negative binomial coefficient
estimates (absolute t-statistics) on the other covariates in the disaggregate model are as follows: Fiber: M).03 (-3.93 ***); Population: 1.56E-08 (0.46); Unemployment: 0.06 (2.62 **); Link-Up: 16.16 (2.96 ***); Wage: 0.07 (3.71 ***); Population Density: -0.01 (3.09 ***); Income: 0.60 (16.72 ***); Rate of Return: 0.25 (2.08 **); Price Cap: 0.53 (5.12 ***); and Rate Moratorium: 0.84 (1.97 *), where * denotes statistical significance at the 10% level, ** statistical significance at the 5% level, and *** statistical significance at the 1% level in a two-tailed test.
Our computation using the binomial coefficient
can become forbidding even if the population size is modestly large.
f.sub.B of (x) = [the binomial coefficient
of n over x] times p.sup.x time (1 minus p).sup.(n-x) where x = 0,1,2,..., n
Notice that if we use the general binomial coefficient
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then (4) can be rewritten as