István Mező

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.

Some physical applications of generalized Lambert functions

In this paper we review the physical applications of the generalized Lambert function recently defined by the first author. Among these applications we mention the eigenstate anomaly of the

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

H_2^+

Real zeros and partitions without singleton blocks

ion, the two dimensional two-body problem in general relativity, the stability analysis of delay differential equations and water-wave applications. We also point out that the inverse Langevin function is nothing else but a specially parametrized generalized Lambert function.

Radii of starlikeness and convexity of some q-Bessel functions

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.

The linear algebra of the r-Whitney matrices

We prove that the generating polynomials of partitions of an n -element set into non-singleton blocks, counted by the number of blocks, have real roots only and we study the asymptotic behavior of the leftmost roots. We apply this information to find the most likely number of blocks. Also, we present a quick way to prove the corresponding statement for cycles of permutations in which each cycle is longer than a given integer r .

The estimation of the zeros of the Bell and r-Bell polynomials

Abstract Geometric properties of the Jackson and Hahn–Exton q -Bessel functions are studied. For each of them, three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane. For each of the six functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q -generalizations of some known results for Bessel functions of the first kind. The characterization of entire functions from the Laguerre–Polya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn–Exton q -Bessel functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q -Bessel function and its derivatives. Some open problems are proposed at the end of the paper.

The r-derangement numbers

The linear algebraic theory of the Pascal and Vandermonde matrix is well developed by many authors. In the last two decades many interrelations have been discovered between the mentioned matrices, their generalizations and the Stirling matrices. We follow this direction and discover new matricial relations by using the so-called r-Whitney numbers. Along this way, we develop two natural extensions of the Vandermonde matrix with which we can study and evaluate successive power sums of arithmetic progressions and win new identities for the r-Whitney numbers.

A nonterminating 7F6-series evaluation

It is a classical result that the zeros of the Bell polynomials are real and negative. In this study we deal with the asymptotic growth of the leftmost zeros of the Bell polynomials and generalize the results for the r-Bell polynomials, too. In addition, we offer a heuristic approach for the approximation of the maximizing index of the Stirling numbers of both kind.

Integrals and derivatives connected to the r-Lambert function

Abstract The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, p -adic valuations, and diophantine problems.