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Robert Frontczak

Publications and source records attributed to Robert Frontczak.

At least 19 recordsLinked to original sources
arXiv2026

Harmonic Catalan Number Series via Half-Integer Binomial Coefficients

We develop a method for evaluating finite and infinite series involving Catalan numbers by specializing alternating binomial identities at half-integer parameters. The key ingredient is an explicit representation of the generalized binomial coefficients $\binom{m+\frac12}{k+1}$ in terms of Catalan numbers and products of odd linear factors. Combining this representation with Pascal-type identities and two identities of Bat{\i}r and Sofo, we derive several families of closed-form summation formulas involving Catalan numbers, harmonic numbers, and odd harmonic numbers. As special cases, we obtain evaluations of finite sums together with their corresponding infinite-series analogues. The approach provides a systematic mechanism for transforming identities for generalized binomial coefficients into identities for Catalan-number series.

math.CO↗
arXiv2026

Abel-Type Transformations and Telescoping Structures in Reciprocal Series of Second-Order Linear Recurrences

We develop a unified method for transforming and evaluating infinite series involving products of terms of second-order linear recurrences in the denominator. The approach is based on a discrete Abel-type summation formula (summation by parts), which converts reciprocal series with three or more factors into expressions exhibiting a partial telescoping structure. As a consequence, we obtain general transformation formulas for series of the form $\sum\limits_{k=1}^{\infty} \frac{(\pm 1)^k}{w_{rk+l} w_{mk+s} w_{m(k+1)+s}}$, together with extensions to products of four or more terms. These formulas provide a systematic framework that unifies and extends many known identities for Fibonacci and Lucas numbers. In addition, the method leads to explicit evaluations and identities involving several classical combinatorial sequences, including Catalan numbers, harmonic numbers, and Stirling numbers of both kinds. A key feature of the approach is that it naturally distinguishes between even and odd values of the parameter $m$, leading to structurally different representations. The results show that summation by parts is an effective and flexible tool for reducing multi-factor reciprocal sums to simpler forms.

math.GM↗
arXiv2026

A combinatorial sum with two complex parameters

This article deals with combinatorial identities with two complex parameters. Starting with a fundamental lemma, we derive various polynomial identities, combinatorial sums and related results. For example, we generalize a polynomial identity of Carlitz involving central binomial coefficients and present a second identity of the same nature. Special cases of our findings lead to sums involving Catalan numbers, harmonic numbers, and Fibonacci numbers.

math.GM↗
arXiv2026

New sums mixing harmonic numbers and central binomial coefficients

We study two new classes of sums with inverse binomial coefficients and harmonic numbers. In addition we establish recursive solutions to the following power sums \begin{equation*} U_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}} \cdot k^d \quad \mbox{and}\quad V_d(n) = \sum_{k=1}^n \frac{2^{2k}}{\binom{2k}{k}}\cdot k^d\,H_k, \end{equation*} where $d$ is a positive integer.

math.NT↗
arXiv2026

Applications of an identity of Bat{\i}r

Based on an interesting identity of Bat{\i}r we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.

math.CO↗
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

Fibonacci-harmonic sums

We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial transformation.

math.GM↗
arXiv2025

On Some Series Involving the Central Binomial Coefficients

In this paper, we explore a variety of series involving the central binomial coefficients, highlighting their structural properties and connections to other mathematical objects. Specifically, we derive new closed-form representations and examine the convergence properties of infinite series with a repeating alternation pattern of signs involving central binomial coefficients. More concretely, we derive the series $$\sum\limits_{n=0}^{\infty}\frac{(-1)^{\omega_n}}{2n+1}\tbinom{2n}{n}x^n,\,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{\omega_n}}\tbinom{2n}{n}x^n\,\,\, \text{and} \,\,\, \sum\limits_{n=0}^{\infty}{(-1)^{\omega_n}}n\tbinom{2n}{n}x^n,$$ where $\omega_n$ represents both $\lfloor\frac{n}{2}\rfloor$ and $\lceil\frac{n}{2}\rceil$. Also, we present novel series involving Fibonacci and Lucas numbers, deriving many interesting identities.

math.CO↗
arXiv2025

Three combinatorial sums involving central binomial coefficients

We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive expressions for these sums. In addition, we offer an alternative approach to express one class of sums and some related sums in closed form in terms of Stirling numbers and r-Stirling numbers of the second kind.

math.NT↗
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

New harmonic number series

Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein.

math.NT↗
arXiv2024

Series associated with a forgotten identity of N\"orlund

We apply a seemingly forgotten series expression of N\"orlund for the psi function to express infinite series involving inverse factorials in closed form. Many of such series contain products of Catalan numbers and (odd) harmonic numbers. We also prove some new series for $\pi$.

math.GM↗
arXiv2024

Convolutions of second order sequences: A direct approach

Using a direct algebraic approach we derive convolution identities for second order sequences, hereby distinguishing between sequences obeying the same or different recurrence relations. We also state a general convolution for Horadam sequences. Convolutions for Chebyshev polynomials will also be stated.

math.GM↗
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↗