In this work, we introduce a symmetric algorithm obtained by the recurrence relation a k n = a k n−1 +a k−1 n . We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacciand Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci-and Lucas numbers are obtained.Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci-and Lucas numbers are investigated.

In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. We show that harmonic versions of these polynomials and their generalizations are useful for obtaining closed forms of some series related to harmonic numbers.

We show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. These results enable us to reformulate Euler's formula involving the Hurwitz zeta function. In additon, we improve Conway and Guy's formula for hyperharmonic numbers.

The hyperharmonic numbers h (r) n are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers:can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil. We also provide an explicit evaluation of σ (r, m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.

In this paper we collect two generalizations of harmonic numbers (namely generalized harmonic numbers and hyperharmonic numbers) under one roof. Recursion relations, closed-form evaluations, and generating functions of this unified extension are obtained. In light of this notion we evaluate some particular values of Euler sums in terms of odd zeta values. We also consider the noninteger property and some arithmetical aspects of this unified extension.