Searcharxiv⌕ Search

arXiv subjects

Kunle Adegoke

Publications and source records attributed to Kunle Adegoke.

At least 73 records · Page 4Linked to original sources
arXiv2019

A master identity for Horadam numbers

We derive an identity involving Horadam numbers. Numerous new identities as well as those found in the existing literature are subsumed in this single identity.

math.NT↗
arXiv2019

An Identity for Second Order Sequences Obeying the Same Recurrence Relation

We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are developed. Illustrative examples are drawn from Fibonacci, Fibonacci-Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences and their generalizations. Our new results subsume previously known identities.

math.GM↗
arXiv2018

Squares of Fibonacci-Like Numbers

We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.

math.GM↗
arXiv2018

New Tribonacci Recurrence Relations and Addition Formulas

Only one three-term recurrence relation, namely, $W_{r}=2W_{r-1}-W_{r-4}$, is known for the generalized Tribonacci numbers, $W_r$, $r\in\mathbb{Z}$, defined by $W_{r}=W_{r-1}+W_{r-2}+W_{r-3}$ and \mbox{$W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}$}, where $W_0$, $W_1$ and $W_2$ are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is, $W_{r + s} = T_{s - 1} W_{r - 1} + (T_{s - 1} + T_{s-2} )W_r + T_s W_{r + 1}$, where $({T_r})_{r\in\mathbb{Z}}$ is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with $W_0=T_0=0$ and $W_1=W_2=T_1=T_2=1$. In this paper we discover three new three-term recurrence relations and two identities from which a plethora of new addition formulas for the generalized Tribonacci numbers may be discovered. We obtain a simple relation connecting the Tribonacci numbers and the Tribonacci-Lucas numbers. Finally, we derive quadratic and cubic recurrence relations for the generalized Tribonacci numbers.

math.CO↗
arXiv2018

Fibonacci and Lucas Identities the Golden Way

By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio, $α=(1+\sqrt 5)/2$ and its inverse, $β=-1/α=(1-\sqrt 5)/2$, a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, we follow the reverse course: we derive numerous Fibonacci and Lucas identities by making use of the well-known expressions for the powers of $α$ and $β$ in terms of Fibonacci and Lucas numbers.

math.NT↗
arXiv2018

A new Fibonacci identity and its associated summation identities

We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also derive several binomial and ordinary summation identities arising from this identity; in particular we obtain a generalization of Halton's general Fibonacci identity.

math.CO↗
arXiv2018

Linear Properties of Generalized $n$-step Fibonacci Numbers

We present numerous interesting, mostly new, results involving the $n$-step Fibonacci numbers and $n$-step Lucas numbers and a generalization. Properties considered include recurrence relations, summation identities, including binomial and double binomial summation identities, partial sums and ordinary generating functions. Explicit examples are given for small $n$ values.

math.NT↗
arXiv2018

Identities for second order recurrence sequences

We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a unified study of six well known integer sequences, namely the Fibonacci sequence, the sequence of Lucas numbers, the Jacobsthal sequence, the Jacobsthal-Lucas sequence, the Pell sequence and the Pell-Lucas sequence.

math.GM↗
arXiv2018

Squares of Tribonacci numbers

We prove some identities for the squares of generalized Tribonacci numbers. Various summation identities involving these numbers are derived.

math.CO↗
arXiv2018

Weighted generalized Fibonacci sums

We derive weighted sums, including binomial and double binomial sums, for the generalized Fibonacci sequence $\{G_m\}$ where for $m\ge 2$, $G_m=G_{m-1}+G_{m-2}$ with initial values $G_0$ and $G_1$.

math.CA↗
arXiv2018

Weighted Tribonacci sums

We derive various weighted summation identities, including binomial and double binomial identities, for Tribonacci numbers. Our results contain some previously known results as special cases.

math.CA↗
arXiv2018

Weighted sums of some second-order sequences

We derive weighted summation identities involving the second order recurrence sequence $\{w_n\} =\{ w_n(a,b; p, q)\}$ defined by $w_0 = a,\,w_1 = b;\,w_n = pw_{n - 1} - qw_{n - 2}\, (n \ge 2)$, where $a$, $b$, $p$ and $q$ are arbitrary complex numbers, with $p\ne 0$ and $q\ne 0$.

math.NT↗
arXiv2017

Sums of fourth powers of Fibonacci and Lucas numbers

We obtain closed-form expressions for all sums of the form \mbox{$\sum_{k = 1}^n {F_{mk}{}^4 }$} and \mbox{$\sum_{k = 1}^n {L_{mk}{}^4 }$} and their alternating versions, where $F_i$ and $L_i$ denote Fibonacci and Lucas numbers respectively. Our results complement those of Melham who studied the alternating sums.

math.NT↗
arXiv2017

Quantization of the one-dimensional free harmonic oscillator as an example of the application of the node theorem and the MacDonald-Hylleraas-Undheim theorem

Using heuristic arguments alone, based on the properties of the wavefunctions, we obtain the energy eigenvalues and the corresponding eigenfunctions of the one-dimensional harmonic oscillator. This approach is considerably simpler and is perhaps more intuitive than the traditional methods of solving a differential equation and manipulating operators.

quant-ph↗