arXiv subjects
Publications and source records attributed to Kunle Adegoke.
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.
In this note, we show how a combinatorial identity of Frisch can be applied to prove and generalize some well-known identities involving harmonic numbers. We also present some combinatorial identities involving odd harmonic numbers which can be inferred straightforwardly from our results.
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.
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.
We evaluate in closed form several alternating infinite series involving the binomial coefficients $C(4n,2n)$ and $C(4n+2,2n+1)$ in the denominator. One of our results generalizes an identity that was obtained experimentally by Sprugnoli in 2006.
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.
Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We derive formulas for $\sum_{j=1}^n{G_{j + t}^6}$ and $\sum_{j=1}^n{(-1)^{j - 1}G_{j + t}^5(G_{j + t - 1} + G_{j + t + 1})}$, thereby extending the results of Ohtsuka and Nakamura who found simple formulas for $\sum_{j=1}^n{F_j^6}$ and $\sum_{j=1}^n{L_j^6}$, where $F_k$ and $L_k$ are the $k$th Fibonacci and Lucas numbers. We also evaluate $\sum_{j = 1}^n {G_{j + t}^3 G_{j + t + 1}^3 } $ and $\sum_{j = 1}^n {G_{j + t - 1}^2 G_{j + t} G_{j + t + 1} G_{j + t + 2}^2 } $, of which the results of Treeby are particular cases.
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.
We present a differential-calculus-based method which allows one to derive more identities from {\it any} given Fibonacci-Lucas identity containing a finite number of terms and having at least one free index. The method has two {\it independent} components. The first component allows new identities to be obtained directly from an existing identity while the second yields a generalization of the existing identity. The strength of the first component is that no additional information is required about the given original identity. We illustrate the method by providing new generalizations of some well-known identities such as d'Ocagne identity, Candido's identity, Gelin-Cesàro identity and Catalan's identity. The method readily extends to a generalized Fibonacci sequence.
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.
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.
Let $F_k$ be the $k$th Fibonacci number. Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We employ the Gerin-Cesàro identity and an identity of Brousseau to evaluate the following sums: $\sum_{j=1}^n{(\pm 1)^{j - 1}G_j^5}$, $\sum_{j = 1}^n {G_{j - 1} G_{j} G_{j + 1} G_{j + 2} G_{j + m} }$, $\sum_{j = 1}^n {(-F_{m - 3})^{n - j} ( F_{m + 2} )^j G_{j - 1} G_{j} G_{j + 1} G_{j + 2} G_{j + m} }$, and $\sum_{j = 1}^n {(-F_{m + 2})^{n - 2}F_{m - 3}^jG_{j + m} \left( {G_{j - 2} G_{j - 1} G_j G_{j + 1} G_{j + 2} G_{j + 3} } \right)^{ - 1} }$. Among other results, we evaluate the sum and alternating sum of products of five consectutive Fibonacci-like numbers, namely $\sum_{j = 1}^n {\left( { \pm 1} \right)^{j - 1} G_j G_{j + 1} G_{j + 2} G_{j + 3} G_{j + 4} }$.
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.
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.
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 $\binom{3n}{n}$.
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.
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.
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.