Beginning with the negative binomial coefficients
, we again find that greater cultural distance and existing immigrant stocks have negative and positive effects, respectively, on the level of the predicted immigrant stock.
In this section we provide a formal definition of the two circulant determinant sequences with binomial coefficients
and derive the formula for their respective n-th term.
For the sake of completeness we give, without proof, two particular theorems in regard to single binomial coefficients
and their integral representations.
1) is replaced here by the central binomial coefficient
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In this section we present some basic notions on words and binomial coefficients
kxn], we obtain the following corollary that will be a key step in connecting the poset binomial coefficients
Just as with ordinary binomial coefficients
, q-binomial coefficients
have multiple combinatorial interpretations and satisfy many identities, a few of which are highlighted below.
denote the usual q-analogs of numbers, factorials and binomial coefficients
Observe that the left- and right-hand sides of Equation (10) are nonzero for finitely many k and that the sum over j is actually finite by the definition of binomial coefficients
We do not require that u and v be coprime, so such sequences need not contain all of the binomial coefficients
in which a geometric line would intersect Pascal's triangle, e.
Keywords: Hankel determinants, binomial coefficients
, almost product form evaluations, differential equations, [gamma]-operators.
Finally, if d [greater than or equal to] 3 and d does not divide k, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the first q binomial coefficient
vanishes for k [not equivalent to] 1 mod d, and the second one vanishes for k [equivalent to] 1 mod d.