One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by $${\left(\frac{2}{\lambda e^t+1}\right)}^{\alpha }{e}^{xt}=\sum _{n=0}^{\infty }{ℰ}_n^{\left(\alpha \right)}(x;\lambda )\frac{t^n}{n!},\lambda \in ℂ\setminus \{-1\},$$ and as an “exceptional family”...

Let $S$ be a fixed symmetric finite subset of $S{L}_d\left({𝒪}_K\right)$ that generates a Zariski dense subgroup of $S{L}_d\left({𝒪}_K\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({𝒪}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.

We note that every positive integer N has a representation as a sum of distinct members of the sequence ${d(n!)}_{n\ge 1}$, where d(m) is the number of divisors of m. When N is a member of a binary recurrence $u={uₙ}_{n\ge 1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{U_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)/\left(U_{d^k}\right))$ (i=1,...,m) or ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{V_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)\left(V_{d^k}\right)$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent....

We present a constructive procedure for establishing explicit formulas of the constituents matrices. Our approach is based on the tools and techniques from the theory of generalized Fibonacci sequences. Some connections with other results are supplied. Furthermore,we manage to provide tractable expressions for the matrix functions, and for illustration purposes we establish compact formulas for both the matrix logarithm and the matrix pth root. Some examples are also provided.

There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set ℱ(N) = {s/r : r,s ∈ ℤ, gcd(r,s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r,m) = 1 are considered).