arXiv subjects

Karol Gryszka

Publications and source records attributed to Karol Gryszka.

5 recordsLinked to original sources

arXiv2025

Double sums associated with binomial transforms

In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new identities for double sums involving special numbers like Bernoulli numbers, Fibonacci numbers, harmonic numbers, Catalan numbers and Stirling numbers of the second kind. We also consider families of polynomials like Fibonacci polynomials, Chebyshev polynomials, Bernoulli polynomials, and others. Finally, we state new double sums involving hyperbolic functions.

math.CO↗

arXiv2025

Double sums involving binomial coefficients and special numbers

In this paper, we find an elementary approach for double sums where the inner sum is binomial but incomplete. We apply our core identity and its relatives to double sums involving famous numbers such as harmonic numbers, Fibonacci numbers, Stirling numbers and $r$-Stirling numbers of the second kind.

math.CO↗

arXiv2025

Finite sums associated with some polynomial identities

In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and also harmonic numbers and squared harmonic numbers. We apply the framework to discuss combinatorial sums associated with some prominent polynomial identities from the recent past.

math.CO↗

arXiv2024

General convolution sums involving Fibonacci m-step numbers

In this paper, using a generating function approach, we derive several new convolution sum identities involving Fibonacci m-step numbers. As special instances of the results derived herein, we will get many new and known results involving Fibonacci, Tribonacci, Tetranacci and Pentanacci numbers. In addition, we establish some general results providing insights into the inner structure of such convolutions. Finally, some mixed convolutions involving Fibonacci m-step numbers and Jacobsthal and Pell numbers will be stated.

math.GM↗

arXiv2020

On the $L^1$ and pointwise divergence of continuous functions

For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.

math.CA↗