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Robert Frontczak

Publications and source records attributed to Robert Frontczak.

At least 37 records · Page 2Linked to original sources
arXiv2024

Combinatorial identities with multiple harmonic-like numbers

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are immediate consequences of the main result. Finally, combinatorial identities involving harmonic-like numbers and other prominent sequences like hyperharmonic numbers and odd harmonic numbers are offered.

math.NT↗
arXiv2024

Integration Formulas Involving Fibonacci and Lucas Numbers

We present a range of difficult integration formulas involving Fibonacci and Lucas numbers and trigonometric functions. These formulas are often expressed in terms of special functions like the dilogarithm and Clausen's function. We also prove complements of integral identities of Dilcher (2000) and Stewart (2022). Many of our results are based on a fundamental lemma dealing with differentiation of complex-valued Fibonacci (Lucas) functions.

math.GM↗
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↗
arXiv2024

Combinatorial sums, series and integrals involving odd harmonic numbers

We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these generating functions we evaluate several infinite series in closed form. In addition, we offer some combinatorial sum identities involving Catalan numbers, harmonic numbers and odd harmonic numbers. Finally, we analyze a special log-integral with Fibonacci numbers and odd harmonic numbers.

math.CO↗
arXiv2023

Some notes on a Fibonacci-Lucas identity

In 2016, Edgar and, independently of him, Bhatnagar sta\-ted a nice polynomial identity that connects Fibonacci and Lucas numbers. Shortly after their publications, this identity has been generalized in two different ways: Dafnis, Phillipou and Livieris provided a generalization to Fibonacci sequences of order $k$ and Abd-Elhameed and Zeyada extended Edgar--Bhatnagar identity to generalized Fibonacci and Lucas sequences. In this paper, we present more polynomial identities for generalized Lucas sequences. We discuss interesting aspects and special cases which have not been stated before but deserve recognition. Finally, we prove the polynomial analogues of these identities for Chebyshev polynomials.

math.CO↗
arXiv2023

Fibonacci sums and divisibility properties

Based on a variant of Sury's polynomial identity we derive new expressions for various finite Fibonacci (Lucas) sums. We extend the results to Fibonacci and Chebyshev polynomials, and also to Horadam sequences. In addition to deriving sum relations, the main identities will be shown to be very useful in establishing and discovering divisibility properties of Fibonacci and Lucas numbers.

math.NT↗
arXiv2023

Fibonacci sums modulo 5

We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.

math.CO↗
arXiv2023

Binomial sum relations involving Fibonacci and Lucas numbers

In this paper, we introduce relations between binomial sums involving (generalized) Fibonacci and Lucas numbers, and different kinds of binomial coefficients. We also present some relations between sums with two and three binomial coefficients. In the course of exploration we rediscover a few relations presented as problem proposals.

math.CO↗
arXiv2023

Binomial Fibonacci sums from Chebyshev polynomials

We explore new types of binomial sums with Fibonacci and Lucas numbers. The binomial coefficients under consideration are $\frac{n}{n+k}\binom{n+k}{n-k}$ and $\frac{k}{n+k}\binom{n+k}{n-k}$. The identities are derived by relating the underlying sums to Chebyshev polynomials. Finally, some combinatorial sums are studied and a connection to a recent paper by Chu and Guo from 2022 is derived.

math.CO↗
arXiv2023

On some series involving the binomial coefficients $\binom{3n}{n}$

Using a simple transformation, we obtain much simpler forms for some series involving binomial coefficients $\binom{3n}n$ derived by Necdet Batir. New evaluations are given; and connections with Fibonacci numbers and the golden ratio are established. Finally, we derive some Fibonacci and Lucas series involving the reciprocals of $\binom{3n}{n}$.

math.CO↗
arXiv2023

A series of Ramanujan, two-term dilogarithm identities and some Lucas series

We study an elementary series that can be considered a relative of a series studied by Ramanujan in Part 1 of his Lost Notebooks. We derive a closed form for this series in terms of the inverse hyperbolic arctangent and the polylogarithm. Special cases will follow in terms of the Riemann zeta and the alternating Riemann zeta function. In addition, some trigonometric series will be expressed in terms of the Clausen functions. Finally, a range of new two-term dilogarithm identities will be proved and some difficult series involving Lucas numbers will be evaluated in closed form.

math.NT↗
arXiv2023

On a problem of Mező and its generalizations to three classes of rational zeta series

We evaluate in closed form three special classes of alternating zeta series with one and two additional parameters. Two classes are expressed as linear combinations of polylogarithms while for the third class we prove an expression involving the incomplete gamma function and the exponential integral. We also present some related series that can be deduced from the main results as well as some series with Fibonacci and Lucas numbers as coefficients. Particular cases of the series presented here will be rediscoveries of identities established by Zhang and Williams, Choi and Srivastava, and Orr, among others. We will also rediscover a series identity published by Mező in 2015 as a problem proposal in the American Mathematical Monthly.

math.CO↗
arXiv2023

Series and sums involving the floor function

Let $(a_n)_{n\geq 0}$ be an arbitrary sequence and $(a_{\lfloor n/k \rfloor})_{n\geq 0}$ its dual floor sequence. We study infinite series and finite generalized binomial sums involving $(a_{\lfloor n/k \rfloor})_{n\geq 0}$. As applications we prove a range of new closed form expressions for Fibonacci (Lucas) series and binomial sum identities as particular cases.

math.CO↗
arXiv2022

New binomial Fibonacci sums

We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.

math.CO↗
arXiv2022

Balancing polynomials, Fibonacci numbers and some new series for $π$

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $π$ involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.

math.NT↗