Polynomials Related to Harmonic Numbers and Evaluation of Harmonic Number Series I

In this paper we focus on two new families of polynomials which are connected with exponential polynomials and geometric polynomials. We discuss their generalizations and show that these new families of polynomials and their generalizations are useful to obtain closed forms of some series related to harmonic numbers.Comment: 18 page

Nearly hyperharmonic functions are infima of excessive functions

Let $\mathfrak X$ be a Hunt process on a locally compact space $X$ such that the set $\mathcal E_{\mathfrak X}$ of its Borel measurable excessive functions separates points, every function in $\mathcal E_{\mathfrak X}$ is the supremum of its continuous minorants in $\mathcal E_{\mathfrak X}$ and there are strictly positive continuous functions $v,w\in\mathcal E_{\mathfrak X}$ such that $v/w$ vanishes at infinity. A numerical function $u\ge 0$ on $X$ is said to be nearly hyperharmonic, if $\int^\ast u\circ X_{\tau_V}\,dP^x\le u(x)$ for all $x\in X$ and relatively compact open neighborhoods $V$ of $x$, where $\tau_V$ denotes the exit time of $V$. For every such function $u$, its lower semicontinous regularization $\hat u$ is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that every Borel measurable nearly hyperharmonic function on $X$ is the infimum of its majorants in $E_{\mathfrak X}$. The major novelties of our approach are the following: 1. A quick reduction to the special case, where starting at $x\in X$ with $u(x)<\infty$ the expected number of times the process $\mathfrak X$ visits the set of points $y\in X$, where $\hat u(y):=\liminf_{z\to y} u(z)<u(y)$, is finite. 2. The statement that the integral $\int u\,d\mu$ is the infimum of all integrals $\int w\,d\mu$, $w\in E_{\mathfrak X}$ and $w\ge u$, not only for measures $\mu$ satisfying $\int w\,d\mu<\infty$ for some excessive majorant $w$ of $u$, but also for all finite measures. At the end, the measurability assumption on $u$ is weakened considerably.Comment: The presentation is improved at various places. In particular, the special case is more restrictive and yields a better intuition, there is a new Lemma 3.5 leading to a simplification in the proof of Theorem 3.4, and the reduction to the special case in Section 4 is shortened. Whereas Sections 5 and 6 are not modified, there is a more general Section

On the Harmonic and Hyperharmonic Fibonacci Numbers

In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which is finite sums of reciprocals of Fibonacci numbers. We obtain spectral and Euclidean norms of circulant matrices involving harmonic and hyperharmonic Fibonacci numbers

On the Norms of Circulant and r−r-Circulant Matrices With the Hyperharmonic Fibonacci Numbers

In this paper, we study norms of circulant and $r-$circulant matrices involving harmonic Fibonacci and hyperharmonic Fibonacci numbers. We obtain inequalities by using matrix norms

Extended Bernoulli and Stirling matrices and related combinatorial identities

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.Comment: Accepted for publication in Linear Algebra and its Application