Kálmán Liptai

• PhD
• rector at Eszterházy Károly University

One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer recurrence sequences as rational polynomial linear combinations of Fibonacci numbers.

Let G0=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_0=0$$\end{document} and G1=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackag...

Consider the diophantine equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$ in positive integers $x_1\le x_2$, and $y_1\le y_2$. Each side of the equation is a product of two terms of a given binary recurrence, respectively. In this paper, we prove that the only solution to the title equation is $(x_1,x_2,y_1,y_2)=(1,2,1,1)$. The main novelty...

We prove a new formula for hyper-Fibonacci numbers, F[k] n , using flssions of certain polynomials. The result is a concise description of the entries of the matrix of hyper- Fibonacci numbers.

A balancing problem associated with two integer sequences is introduced. The problem is studied using the sequences obtained from a binary recurrence and its associate sequence. We provide an algorithm to solve, under some circumstances, the Diophantine equation G 0 + G 1 + · · ·+G x = H 0 +H 1 +· · ·+H y in the non-negative integer unknowns x and...

By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their arithmetical and combinatorial properties. As an application, at the end of the paper we present a new infinite series representation of the Riemann zeta function via the Lambert $W$.

By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their arithmetical and combinatorial properties. As an application, at the end of the paper we present a new infinite series representation of the Riemann zeta function via the Lambert $W$.

In Komatsu's work (2013), the concept of poly-Cauchy numbers is introduced as an analogue of that of poly-Bernoulli numbers. Both numbers are extensions of classical Cauchy numbers and Bernoulli numbers, respectively. There are several generalizations of poly-Cauchy numbers, including poly-Cauchy numbers with a q parameter and shifted poly-Cauchy n...

We show some relationships between poly-Cauchy type numbers and poly-Bernoulli type numbers. Poly-Cauchy type numbers include poly-Cauchy numbers introduced by T. Komatsu [Kyushu J. Math. 67, No. 1, 143–153 (2013; Zbl 06204389)], and poly-Bernoulli type numbers include poly-Bernoulli numbers introduced by M. Kaneko [J. Théor. Nombres Bordx. 9, No....

Let a, b be nonnegative coprime integers. We call an integer an+b∈ℕ (denoted by B m (a,b) ) an (a,b)-type balancing number if (a+b)+(2a+b)+⋯+(a(n-1)+b)=(a(n+1)+b)+⋯+(a(n+r)+b) for some r∈ℕ. In this paper we consider and give numerical results for the equation B m (a,b) =f(x) where B m (a,b) is an (a,b)-type balancing number and f(x) is a polynomial...

The Diophantine equation F1k + F2k + ⋯ + Fn-1k = Fn+1l + Fn+2l + ⋯ + Fn+rl in positive integers n,r,k,l with n ≥ 2 is studied where F n is the nth term of the Fibonacci sequence.

The present paper studies the diophantine equation G n H n + c = x 2n and related questions, where the integer binary recurrence sequences {G}, {H} and {x} satisfy the same recurrence relation, and c is a given integer. We prove necessary and sufficient conditions for the solubility of G n H n + c = x 2n . Finally, a few relevant examples are provi...

A positive integer n is called a balancing number if 1+⋯+(n - 1) = (n + 1) + ⋯ + (n + r) for some positive integer r. Balancing numbers and their generalizations have been investigated by several authors, from many aspects. In this paper we introduce the concept of balancing numbers in arithmetic progressions, and prove several e®ective finiteness...

Let R i = R(A, B, R 0, R 1) be a second order linear recurrence sequence. In the present paper we prove that any sequence R i = R(A, B, 0, R1) with D = A 2 + 4B > 0, (A, B) ≠ (0,1) is not a balancing sequence.

The aim of this paper is to investigate the zeros of polynomials P n,k (x)=K k-1 x n +K k x n-1 +⋯+K n+k2 x+K n+k-1 , where the coefficients K i are terms of a linear recursive sequence of k-order (k≥2).

Properties of dispersion of block sequences were investigated by J. T. Tóth, L. Mišík and F. Filip [Math. Slovaca 54, 453–464 (2004; Zbl 1108.11017)]. The present paper is a continuation of the study of relations between the density of the block sequence and so called dispersion of the block sequence.

The positive integer x is a (k, l) -balancing number for y(x ≤ y — 2) if 1k + 2k + … + (x — 1)k = (x + 1)l + … + (y — 1)l for fixed positive integers k and l. In this paper, we prove some effective and ineffective finiteness statements for the balancing numbers, using certain Baker-type Diophantine results and Bilu—Tichy theorem, respectively.

In this article we present the cryptography of the food safety tracking sys-tem of the Regional Knowledge Center (EGERFOOD), which can be found in Eger, Hungary at the Eszterházy Károly College. We analyzed its require-ments for the underlying information system. To build a user friendly system, which serves quickly and cost effectively the costume...

In this paper we study sequences of the form (an + b)1n=1, where a, b 2 N. We prove many interesting results connection with sequences with infinitely many prime divisors.

A linear recursive sequence G of order k is defined by the integer initial terms G 0, G 1, . . . , G k-1, integer constants A 1, A 2, . . ., A k and by the recursion G n = A 1G n-1 + . . .+ A kG n-k for k ≤ n. In the case k = 2, G 0 = 0, G 1 = 1 (when we denote the sequence by R) it is known that there are only finitely many perfect powers in such...

A positive integer n is called a balancing number if 1 + 2 + ⋯ + (n - 1) = (n + 1) + (n + 2) + ⋯ + (n + r) for some natural number r. We prove that there is no Fibonacci balancing number except 1.

Let G (i) ={G (i) x } 1 x=0 (i=1,2,...,m) linear recursive sequences and let F (x)=dx q + dpx p +dp 1x p 1 +��� +d0, where d and di's are rational integers, be a polynomial. In this paper we showed that for the equations m P i=1 G (i) xi =F (x) and m Q i=1 G (i) xi =F (x) where xi-s are non-negative

Let \( \{ {R_n}\} \begin{array}{*{20}{c}} \infty \\ {n = 0} \\ \end{array} \) and \( \{ {V_n}\} \begin{array}{*{20}{c}} \infty \\ {n = 0} \\ \end{array} \) be second order linear recurring sequences of integer defined by $$ {R_n} = A{R_{n - 1}} - B{R_{n - 2}}{\rm{ }}(n > 1) $$,$$ {V_n} = A{V_{n - 1}} - B{V_{n - 2}}{\rm{ }}(n > 1) $$ where A > 0 and...

Let G be a linear recursive sequence of order k satisfying the recursion Gn =A1 Gn−1 +···+Ak Gn−k. In the case k=2 it is known that there are only finitely many perfect powers in such a sequence. Ribenboim and McDaniel proved for sequences with k=2, G0 =0 and G1 =1 that in general for a term Gn there are only finitely many terms Gm such that Gn Gm...

A linear recursive sequence G of order k is defined by the integer initial terms G 0 ,G 1 ,⋯,G k-1 , integer constants A 1 ,A 2 ,⋯,A k and by the recursion G n =A 1 G n-1 +⋯+A k G n-k for k≤n. In the case k=2, it is known that there are generally only finitely many perfect powers in the sequence. T. N. Shorey and C. L. Stewart [J. Number Theory 27,...

Let {R n } n=0 ∞ and {V n } n=0 ∞ (n=0,1,2,⋯) be sequences of integers defined by R n =AR n-1 -BR n-2 and V n =AV n-1 -BV n-2 , where A and B are fixed non-zero integers. We give a condition when the distance from the points P n (R n ,V n ) to the line y=Dx tends to zero. Moreover we show that there is no lattice point (x,y) nearer than P n (R n ,V...

A 2005-2008-as periódusban, azaz az OTKA 4 éve alatt a kutatócsoportunk a szerződésben vállalt témák kutatásával foglalkozott. Így, tanulmányoztuk a polinomiális-exponenciális diofantikus egyenleteket, a lineáris rekurziókat, a balansz számokat, az unimodális sorozatokat és a lineáris rekurziókhoz kapcsolódó polinomsorzatok polinomjai gyökeit és az...