an:05990964
Zbl 1249.11032
Liu, Guodong
The \(D\) numbers and the central factorial numbers
EN
Publ. Math. 79, No. 1-2, 41-53 (2011).
00292308
2011
j
11B68 11B83 05A19
Bernoulli polynomials; Bernoulli numbers; the \(D\) numbers; the \(D\) numbers of the second kind; central factorial numbers of the first kind; central factorial numbers of the second kind
The so-called \(D\)-numbers of the first kind of order \(k\) are defined by the generating function
\[
(t\csc t)^k=\sum_{n=0}^\infty(-1)^nD_{2n}^{(k)}\frac{t^{2n}}{(2n)!}.
\]
The \(D\)-numbers of the second kind may be defined by
\[
\frac{t}{\log(t+\sqrt{1+t^2})}=\sum_{n=0}^\infty d_{2n}t^{2n}.
\]
The author connects these numbers and the central factorial coefficients. Latter numbers can be defined by polynomial identities. More precisely, the central factorial coefficients \(t(n,k)\) of the first kind are the coefficients of the polynomial
\[
x\left(x+\frac{n}{2}-1\right)\cdots\left(x+\frac{n}{2}-n+1\right),
\]
while the central factorial coefficients \(T(n,k)\) of the second kind are the linear combination coefficients of \(x^n\) with respect to the base
\[
x\left(x+\frac{k}{2}-1\right)\cdots\left(x+\frac{k}{2}-k+1\right).
\]
The author proves a number of identities with respect to these numbers. Some of them connect these numbers to the classical Bernoulli numbers.
Istv??n Mez?? (Debrecen)