Motivated by a problem on the Maclaurin expansion of \arctan \sqrt{2e^{-z}-1}/\sqrt{2e^{-z}-1}\ , the authors derive higher-order derivatives and series expansions of arctan z/z and arctan√ z/√z , express them via the Gauss hypergeometric functions {}_{2}F_{1}\ and the Chebyshev polynomials Un(x), and determine their asymptotics as n → ∞. Several new identities follow, including a trigonometric identity and an infinite series identity from two Fourier–Legendre expansions of the elliptic integral K (x). Alternative proofs of known {}_{2}F_{1}\ closed forms are also given.
This corrigendum repairs the auxiliary construction used in the proof of Theorem 7 and corrects some related points in Theorems 5, 6, 8, and 9 and Corollary 2.
In this study, a generalization of the Hermite based second kind Peters-type Simsek polynomials is introduced. By means of the generating function, an explicit formula, summation formulas, a recurrence relation and a determinant representation are obtained. Using differential and integral operators, relations with the Stirling, Cauchy and Peters-type Simsek numbers are established. Two subfamilies, namely the three-variable and the parametric ones, are also defined, and their graphs and real roots are analyzed.
In this manuscript, we investigate a linear operator Tg acting on complex polynomials, defined through the Schur-Szegő composition theorem. For this operator, we establish sharp Bernstein-type inequalities for polynomials on the unit disk. Our results not only recover several known theorems as special cases but also provide new estimates for wider classes of Bn-Operators. This work strengthens the link between operator theory and polynomial inequalities, with applications to geometric function theory and approximation theory.
In this paper we investigate the behavior of crossing number under a graph transformation, called ΔY-move, on the complete graph \mathrm{K_n} . Concretely it is shown that for any k ∈ N, there exist a natural number n and a sequence of ΔY-moves \mathrm{K_n} → G^{(1)} → ...→ G^{(k)} which is strictly decreasing with respect to the crossing number, that is, cr( \mathrm{K_n} ) > cr( G^{(1)} ) >... > cr( G^{(k)} ) . We also discuss the decrease of crossing number for n = 6, 7.
