On convexity and power series expansion for logarithm of normalized tail of power series expansion for square of tangent (Q6618759)

Given a real function \(f(x)\) with formal Maclaurin expansion \(\sum_{k=0}^{\infty} f^{(k)}(0) \frac{x^{k}} {k!}\), if \(f^{(n+1)}(0)\neq0 \) for some nonnegative integer number \(n,\) the generalized tail of \(f(x)\) of order \(n\) is the function \(f_{n}(x)=\frac{1}{ f^{(n+1)}(0)} \frac{(n+1)!}{x^{n+1}} \big[ f(x) - \sum_{k=0}^{n} f^{(k)}(0) \frac{x^k}{k!}\big], x\neq0, f_{n}(0)=1.\) For the exponential function such a concept has been considered in [\textit{Z.-H. Bao} et al., Symmetry 16, No. 8, Article ID 989, 15 p. (2024; \url{doi:10.3390/sym16080989})], and recently studied by several authors for other special functions.\N\NIn [\textit{Yu. A. Brychkov}, Integral Transforms Spec. Funct. 20, No. 11--12, 797--804 (2009; Zbl 1183.33010)]4, the Maclaurin power series expansion of the function \(f(x)= tan^2 x\) is deduced. Indeed, it is given as \(\sum_{j=1}^{\infty} \frac{2^{2j+2} (2^{2j+2}-1) (2j+1)} {(2j+2)!} |B_{2j+2}| x^{2j}, |x|< \frac{\pi}{2},\) where \(B_{k}\) is the \(k\)-th Bernoulli number. For every \(n\geq 1\) its normalized tail of order \(2n-1\) is a positive and increasing function on \((0, \frac{\pi}{2}).\) As a consequence, the function \(H_{n}(x)= ln f_{2n-1}(x), n\geq 1,\) is defined and even on \((- \frac{\pi}{2} , \frac{\pi}{2}),\) decreasing on \( (- \frac{\pi}{2} , 0)\) and increasing on \((0 , \frac{\pi}{2}).\)\N\NIn the paper under review, the authors prove that for every \(n\geq1\), the function \(H_{n}(x)\) is convex on \((-\frac{\pi}{2} , \frac{\pi}{2}).\) On the other hand, the Maclaurin power series expansion around \(x=0\) of \(H_{n}(x), n\geq 1,\) is obtained.