Formulas for power of the hyperbolic tangent with an application to higher-order tangent numbers (Q812500)

The tangent numbers \(\{C_{2n+1}\}_{n=0}^{\infty}\subset \mathbb{Z}\) are defined by \[ \tanh(t)=-\sum_{n=0}^{\infty}C_{2n+1}\frac{t^{2n+1}}{(2n+1)!}\,. \] The higher-order tangent numbers \(C_{2k+n}^{(n)},k=0,1,2\ldots \) and \(n=1,2,3, \ldots\), are defined by \[ \tanh^{n}(t)=(-1)^{n}\,\sum_{k=0}^{\infty}C_{2k+n}^{(n)}\,\frac{t^{2k+n}}{(2k+n)!}\,. \] The author shows that the function \(\tanh^{2n+1}(x)\) is a linear combination of even-order derivatives of \(\tanh(x)\), while the function \(1-\tanh^{2n+2}(x)\) is a linear combination of odd-order derivatives of \(\tanh(x)\). These results are then used to express higher-order tangent numbers as linear combinations of the ordinary tangent numbers.