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 [12] 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.
