Lucas' theorem: its generalizations, extensions and applications (1878--2014)

In 1878 \'E. Lucas proved a remarkable result which provides a simple way to compute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms of the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the $p$-adic expansions of nonnegative integers $n$ and $m$, then \begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}. \end{equation*}} The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of Lucas}), plays an important role in Number Theory and Combinatorics. In this article, consisting of six sections, we provide a historical survey of Lucas type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas like theorems for some generalized binomial coefficients, and some their applications. In Section 1 we present the fundamental congruences modulo a prime including the famous Lucas' theorem. In Section 2 we mention several known proofs and some consequences of Lucas' theorem. In Section 3 we present a number of extensions and variations of Lucas' theorem modulo prime powers. In Section 4 we consider the notions of the Lucas property and the double Lucas property, where we also present numerous integer sequences satisfying one of these properties or a certain Lucas type congruence. In Section 5 we collect several known Lucas type congruences for some generalized binomial coefficients. In particular, this concerns the Fibonomial coefficients, the Lucas $u$-nomial coefficients, the Gaussian $q$-nomial coefficients and their generalizations. Finally, some applications of Lucas' theorem in Number Theory and Combinatorics are given in Section 6.Comment: 51 pages; survey article on Lucas type congruences closely related to Lucas' theore

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials'$ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial$ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya$ and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure

Divisors and specializations of Lucas polynomials

Three-term recurrences have infused stupendous amount of research in a broad spectrum of the sciences, such as orthogonal polynomials (in special functions) and lattice paths (in enumerative combinatorics). Among these are the Lucas polynomials, which have seen a recent true revival. In this paper one of the themes of investigation is the specialization to the Pell and Delannoy numbers. The underpinning motivation comprises primarily of divisibility and symmetry. One of the most remarkable findings is a structural decomposition of the Lucas polynomials into what we term as flat and sharp analogs.Comment: Minor typos are fixed, new references are added. To appear in Journal of Combinatoric

Sobre problemas envolvendo números de k-bonacci e coeficientes fibonomiais

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2017.Os números de Fibonacci possui várias generalizações, entre elas temos a sequência (Fn (k))n que é chamada de sequência de Fibonacci k-generalizada. Observando a identidade F2 n+F2 n+1=F2n+1, Chaves e Marques, em 2014, provaram que a equação Diofantina (Fn (k))2+ (F(k) n+1)2= Fm (k) não possui soluções em inteiros positivos n, m e k, com n > 1 e k ≥ 3. Nesse trabalho, mostramos que a equação Diofantina (Fn (k))2 +(F(k) n+1)2 = Fm (l), não possui solução para 2≤ k k + 1. Outra generalização da sequência de Fibonacci s˜ao os coeficientes fibonomiais. Em 2015, Marques e Trojovský provaram que uma condição mais fraca. se p ≡ ± 1 (mod 5), então p † [pa+1 pa] , para todo a ≥ 1.Nesse trabalho, encontramos as classe de resíduos de módulo p, p2, p3 e p4, quando p ≡ ± 1 (mod 5) e sobre uma condição mais fraca. Em particular, provamos que se p é um número primo tal que p ≡ ± 1 (mod 5), então [pa+1 pa] ≡ 1 (mod p).Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) e Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES).Regarding the identity F2 n+F2 n+1=F2n+1, Chaves and Marques, in 2014, proved that (Fn (k))2+ (F(k) n+1)2= Fm (k) does not have solution for integers n, m e k, with n > 1 and k ≥ 3. In this work, we show that (Fn (k))2 +(F(k) n+1)2 = Fm (l) does not have solutions for 2≤ k k + 1. Another generalization of the Fibonacci sequence are the Fibonomial coe#cients. In 2015, Marques and Trojovský proved that if p ≡ ± 1 (mod 5), then p † [pa+1 pa] for all a ≥ 1. In this work, we also find the residue class of [pa+1 pa] modulo p, p2, p3 e p4, when p ≡ ± 1 (mod 5) under some weak hypothesis. In particular, we proved that if p is a prime number such that p ≡ ± 1 (mod 5), then [pa+1 pa] ≡ 1 (mod p)

On alternative definition of Lucas atoms and their pp-adic valuations

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in \cite{ST}. The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the $p$-adic valuations of Lucas atoms for any prime $p$, answering to a problem left open in \cite{ST}, where the authors treated only some specific cases for $p \in \{2, 3\}$. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two