Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark–İsmail's two conjectures (pp.443-465)
Y.-F. Li¹, D. Lim^2,*, F. Qi^{1, 3, 4,*}
https://doi.org/10.30546/1683-6154.22.4.2023.443

¹School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454010, Henan, China

²Department of Mathematics Education, Andong National University, Andong 36729, Republic of Korea

³School of Mathematics and Physics, Hulunbuir University, Inner Mongolia 021008, China

⁴Independent researcher, Dallas, TX 75252-8024, USA

^∗Corresponding authors' e-mail:  dklim@anu.ac.kr,  honest.john.china@gmail.com

Abstract. In the paper, by virtue of the famous formula of Faà di Bruno, with the aid of several identities of partial Bell polynomials, by means of a formula for derivatives of the ratio of two differentiable functions, and with availability of other techniques, the authors establish closed-form formulas in terms of the Bernoulli numbers and the second kind Stirling numbers, present determinantal expressions, derive recursive relations, obtain power series, and compute special values of the function v^j/1−e^−v, its derivatives, and related ones used in Clark–Ismail's two conjectures. By these results, the authors also discover a formula for the determinant of a Hessenberg matrix and derive logarithmic convexity of a sequence related to the function and its derivatives.

Keywords: Conjecture, Bell Polynomial of the Second Kind, Faà di Bruno Formula, Closed-Form Formula, Recursive Relation, Special Value, Exponential Function, Positivity, Stirling Number of the Second Kind, Power Series, Hessenberg Determinant, Bernoulli Number, Logarithmic Convexity.

AMS Subject Classification: 15A15, 26A24, 26A48, 26A51, 33B15, 44A10, 41A58.