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Taras Goy

Publications and source records attributed to Taras Goy.

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↗
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↗
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 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

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↗
arXiv2023

Hessenberg-Toeplitz Matrix Determinants with Schroder and Fine Number Entries

In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schroder and Fine number sequences. Algebraic proofs of these results can be given which make use of Trudi's formula and the generating function of the associated sequence of determinants. We also provide direct arguments of our results that utilize various counting techniques, among them sign-changing involutions, on combinatorial structures related to classes of lattice paths enumerated by the Schroder and Fine numbers. As a consequence of our results, we obtain some new formulas for the Schroder and Catalan numbers as well as for some additional sequences from the OEIS in terms of determinants of certain Hessenberg-Toeplitz matrices.

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

Additional Fibonacci-Bernoulli relations

We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations of hyperbolic functions. Special cases and some corollaries will highlight interesting aspects of our findings.

math.CO↗
arXiv2022

Fibonacci-Catalan Series

We study certain series with Catalan numbers and reciprocal Catalan numbers, respectively, and provide seemingly new closed form evaluations of these series with Fibonacci (Lucas) entries. In addition, we state some combinatorial sums that can be inferred from the series.

math.CO↗
arXiv2022

On a family of infinite series with reciprocal Catalan numbers

We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan-Fibonacci relations. Then we focus on the general properties of the family and prove explicit formulas, including two types of integral representations.

math.CO↗