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The \(D\) numbers and the central factorial numbers. (English) Zbl 1249.11032
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.

11B68 Bernoulli and Euler numbers and polynomials
11B83 Special sequences and polynomials
05A19 Combinatorial identities, bijective combinatorics
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