Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine function and the inverse hyperbolic cosine function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

Some properties of the Hermite polynomials

In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers

In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.

Three closed forms for convolved Fibonacci numbers

In the paper, by virtue of the Faa di Bruno formula and several properties of the Bell polynomials of the second kind, the author computes higher order derivatives of the generating function of convolved Fibonacci numbers and, consequently, derives three closed forms for convolved Fibonacci numbers in terms of the falling and rising factorials, the Lah numbers, the signed Stirling numbers of the first kind, and the golden ratio.

Simplifying coefficients in a family of nonlinear ordinary differential equations

By virtue of the Faá di Bruno formula, properties of the Stirling numbers and the Bell polynomials of the second kind, the binomial inversion formula, and other techniques in combinatorial analysis, the author finds a simple, meaningful, and signicant expression for coefficients in a family of nonlinear ordinary differential equations.

Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius-Euler Numbers

In the paper, the authors apply Faà di Bruno formula, some properties of the Bell polynomials of the second kind, the inversion formulas of binomial numbers and the Stirling numbers of the first and the second kind, to significantly simplify coefficients in two families of ordinary differential equations associated with the higher order Frobenius–Euler numbers.

An exact and explicit implied volatility inversion formula

In this paper, we develop an exact and explicit (model-independent) Taylor series representation of the implied volatility based on the novel applications of an extended Faà di Bruno formula under the operator calculus setting, and the Lagrange inversion theorem. We rigorously establish that our formula converges to the true implied volatility as the truncation order increases. Numerical examples illustrate the remarkable accuracy and efficiency of the formula. The formula distinguishes from previous literature as it converges to the true exact implied volatility, is a closed-form formula whose coefficients are explicitly determined and do not involve numerical iterations.

Simple forms for coefficients in two families of ordinary differential equations

In the paper, by virtue of the Faá di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion formula for the Stirling numbers of the first and second kinds, the author finds simple, meaningful, and significant forms for coefficients in two families of ordinary differential equations.