Ayhan Dil

A symmetric algorithm for hyperharmonic and Fibonacci numbers

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

Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.

Geometric polynomials: properties and applications to series with zeta values

We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann’s zeta function.

Series with Hermite polynomials and applications

We obtain a series transformation formula involving the classical Her- mite polynomials. We then provide a number of applications using appropriate binomial transformations. Several of the new series involve Hermite polynomials and harmonic numbers, Lucas sequences, exponential and geometric numbers. We also obtain a series involving both Hermite and Laguerre polynomials, and a series with Hermite polynomi- als and Stirling numbers of the second kind.