a b s t r a c tIn this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.

In this paper, we discuss the log-behavior of the Catalan–Larcombe–French sequence {Pn}n≥0. We prove that {Pn}n≥0 is log-balanced and {Pn/(n!)2}n≥0 is unimodal. In addition, we show that {Pk/k!}0≤k≤n is reverse ultra log-concave.

In this paper, we discuss the properties of a class of generalized harmonic numbers H.n; r/. By means of the method of coefficients, we establish some identities involving H.n; r/. We obtain a pair of inversion formulas. Furthermore, we investigate certain sums related to H.n; r/, and give their asymptotic expansions. In particular, we obtain the asymptotic expansion of certain sums involving H.n; r/ and the inverse of binomial coefficients by Laplace's method.

This paper considers the generalized Stirling numbers of the first and second kinds. First, we show that the sequences of the above generalized Stirling numbers are both log-concave under some mild conditions. Then, we show that some polynomials related to the above generalized Stirling numbers are $q$-log-concave or $q$-log-convex under suitable conditions. We further discuss the log-convexity of some linear transformations related to generalized Stirling numbers of the first kind.

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