Aimin Xu

By utilizing the generating function of higher-order Bernoulli polynomials, we uncover novel relationships that intertwine higher-order Bernoulli polynomials, higher-order Bernoulli numbers, Stirling numbers of the second kind, and central factorial numbers of the second kind. Leveraging these interconnections, we successfully rederive the identities formulated by Qi and Taylor, specifically those pertaining to Stirling numbers of the second kind and central factorial numbers of the second kind. Additionally, we derive series expansions for both positive integer and real powers of the sinc and sinhc functions.

In this paper two kinds of identities involving derangement polynomials and $ r $-Bell polynomials were established. The identities of the first kind extended the identity on derangement numbers and Bell numbers due to Clarke and Sved and its generalizations due to Du and Fonseca. The identities of the second kind extended some of the results on derangement polynomials and Bell polynomials due to Kim et al.

In this paper, we establish a new asymptotic expansion related to the Wallis ratio. By using the exponential Bell polynomials, we show that the coefficients of the asymptotic expansion can be recursively determined. In addition, an explicit expression for the coefficients is given. Our results improve and generalize the existing ones [1].

Several conjectures posed by Qi on completely monotonic degrees of remainders for the asymptotic formulas of the digamma and trigamma functions are proved.

By the LU factorization, we evaluate a determinant involving the complete symmetric functions. From a viewpoint of symmetric functions, some results for the evaluations of the determinants of the matrices consisting of the numbers of the Stirling-type are given.