Power series expansion of Wilf function

TL;DR

This paper derives power series expansions for functions involving inverse hyperbolic tangent and the Wilf function, analyzes their properties, and finds closed-form formulas for special values of hypergeometric and Bell polynomials.

Contribution

It introduces new power series expansions for the Wilf function and related functions, expressing coefficients via Stirling numbers, and explores their properties and applications.

Findings

Power series expansion of the Wilf function is established.

Coefficients are expressed in terms of Stirling numbers of the second kind.

Closed-form formulas for special hypergeometric and Bell polynomial values are derived.

Abstract

In the research, with aid of the Fa\`a di Bruno formula, be virtue of several identities for the Bell polynomials of the second kind, with help of two combinatorial identities, by means of the (logarithmically) complete monotonicity of generating functions of several integer sequences, and in light of the Wronski theorem, the author \begin{enumerate} \item establishes the Taylor power series expansions of several functions involving the inverse (hyperbolic) tangent function; \item finds out the Maclaurin power series expansion of the Wilf function, which is a composite of the inverse tangent, square root, and exponential functions; \item expresses the coefficients in the Maclaurin power series expansion of the Wilf function in terms of the Stirling numbers of the second kind; \item analyzes some properties, including generating functions, limits, positivity, monotonicity, and logarithmic convexity, of the coefficients in the Maclaurin power series expansion of the Wilf function; \item derives a closed-form formula for a sequence of special values of the Gauss hypergeometric function; \item discovers a closed-form formula for a sequence of special values of the Bell polynomials of the second kind; \item presents several infinite series representations of the circular constant and other sequences; \item recovers an asymptotic rational approximation to the circular constant; \item and connects several integer sequences by determinants. \end{enumerate}