In this section, we define two new special power series involving the numbers of Lyndon words and binomial coefficients
. We give relations between these series and zeta-type functions.
Kim  provides another annulus containing all the zeros of a polynomial with the binomial coefficients
where [F.sub.m](n) = n!/[m!(n - m)!] is the binomial coefficient
and i = [square root of -1].
By writing [mathematical expression not reproducible] using a well-known identity of binomial coefficients
and then applying (4), we obtain
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.
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].
We know that [[??].sup.([alpha]).sub.1,l] are the binomial coefficients
in formula (12).
Notice that if we use the general binomial coefficient
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then (4) can be rewritten as
The first is the right-circulant determinant sequence with binomial coefficients
and the second is the left-circulant determinant sequence with binomial coefficients
Equations (2)-(8) involve binomial coefficients
, whose values are large integer numbers for high upper indices, whereas powers of Courant numbers may be very small real numbers.
Among the topics are counting and proofs, algorithms with ciphers, binomial coefficients
and Pascal's triangle, graph traversals, and probability ad expectation.
The proof is completed by induction on k + l and the recursive definition of binomial coefficients