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We give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents of these continued fraction expansions. Linear fractional transformations of such continued fractions are also discussed. We show more continued fraction expansions for different numbers and types, in particular, on Cauchy numbers.

For any positive integer k and prime p, we define an explicit set A k , p of positive integers n, having positive upper and lower density, on which the p-adic valuation of the Stirling number of the first kind s ( n + 1 , k + 1 ) is a nondecreasing function of n and is given exactly by a simple formula. This is motivated by, and extends, some recent results of Lengyel.

We define q -poly-Bernoulli polynomials B n , ρ , q ( k ) ( z ) with a parameter ρ , q -poly-Cauchy polynomials of the first kind c n , ρ , q ( k ) ( z ) and of the second kind c n , ρ , q ( k ) ( z ) with a parameter ρ by Jackson’s integrals, which generalize the previously known numbers and polynomials, including poly-Bernoulli numbers B n ( k ) and the poly-Cauchy numbers of the first kind c n ( k ) and of the second kind c n ( k ) . We investigate their properties connected with usual Stirling numbers and weighted Stirling numbers. We also give the relations between generalized poly-Bernoulli polynomials and two kinds of generalized poly-Cauchy polynomials.

We define poly-Bernoulli numbers and polynomials with a q parameter. These numbers and polynomials show a correspondence to poly-Cauchy numbers and polynomials with a q parameter recently defined by the second author. We study some arithmetical and number theoretical properties of these numbers and polynomials as well.

Here, we find a lower bound for | F n ⋯ F n + k − 1 − F m l | for positive integers k , l , m and n in terms of max { k , l , m , n } , where F s is the s th Fibonacci number.

Extending the work of Burger et al., here we show that every quasi-periodic simple continued fraction α can be transformed into a quasi-periodic non-simple continued fraction having period length one. Moreover, a certain kind of quasi-periodic non-simple continued fraction is equivalent to a quasi-periodic N-continued fraction. The results of this paper follow from arguments of Burger et al. but we apply our version to offer new continued fractions for certain classes of real numbers.

We show some new variations on Tasoev's continued fractions [ 0 ; a k , … , a k ︸ m ¯ ] k = 1 ∞ , where the periodic parts include the exponentials in k instead of the polynomials in k . We also mention some relations with other kinds of continued fractions, in particular, with Rogers–Ramanujan continued fractions.

We consider a new algorithm yielding the values M (θ, φ)=lim inf|q|→∞ |q|‖qθ−φ‖. Specifically, we compute M (θ, 1/a), M (θ, 1/2a), M (θ, 1/ a 2 +4 ) , and M (θ, 1/2) when θ=( a 2 +4 −a)/2=[0, a, a, ...] .

A recent paper (J. Number Theory42(1992), 61–87) announced various arithmetical properties of the Mahler functionf(θ, φ;x, y)=∑∞k=1∑1≤m≤kθ+φxkym. Unfortunately the arguments of that paper are marred by an error whereby the arguments hold only forφ=0 (or whenbn=1 for all positive integersn). We show how to correct the arguments so that they do hold for generalφ. Moreover, another paper (J. Number Theory43(1993), 293–318) soon afterwards happened again to discuss the subject using different expressions for the power series forf. We show that the corrected results here do coincide with those later results notwithstanding our alternative presentation