546 research outputs found

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

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    We deliver here second new H(x)βˆ’binomialsβ€²\textit{H(x)}-binomials' recurrence formula, were H(x)βˆ’binomialsβ€²H(x)-binomials' array is appointed by Wardβˆ’HoradamWard-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,qβˆ’binomialp,q-binomial coefficients onto qβˆ’binomialq-binomial coefficients interpretations thus bringing us back to GyoΒ¨rgyPoˊlyaGy{\"{o}}rgy P\'olya and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure

    Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals

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    Zeckendorf's theorem states that every positive integer can be written uniquely as a sum of non-consecutive Fibonacci numbers Fn{F_n}, with initial terms F1=1,F2=2F_1 = 1, F_2 = 2. We consider the distribution of the number of summands involved in such decompositions. Previous work proved that as nβ†’βˆžn \to \infty the distribution of the number of summands in the Zeckendorf decompositions of m∈[Fn,Fn+1)m \in [F_n, F_{n+1}), appropriately normalized, converges to the standard normal. The proofs crucially used the fact that all integers in [Fn,Fn+1)[F_n, F_{n+1}) share the same potential summands. We generalize these results to subintervals of [Fn,Fn+1)[F_n, F_{n+1}) as nβ†’βˆžn \to \infty; the analysis is significantly more involved here as different integers have different sets of potential summands. Explicitly, fix an integer sequence Ξ±(n)β†’βˆž\alpha(n) \to \infty. As nβ†’βˆžn \to \infty, for almost all m∈[Fn,Fn+1)m \in [F_n, F_{n+1}) the distribution of the number of summands in the Zeckendorf decompositions of integers in the subintervals [m,m+FΞ±(n))[m, m + F_{\alpha(n)}), appropriately normalized, converges to the standard normal. The proof follows by showing that, with probability tending to 11, mm has at least one appropriately located large gap between indices in its decomposition. We then use a correspondence between this interval and [0,FΞ±(n))[0, F_{\alpha(n)}) to obtain the result, since the summands are known to have Gaussian behavior in the latter interval. % We also prove the same result for more general linear recurrences.Comment: Version 1.0, 8 page

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

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    In 1878 \'E. Lucas proved a remarkable result which provides a simple way to compute the binomial coefficient (nm){n\choose m} modulo a prime pp in terms of the binomial coefficients of the base-pp digits of nn and mm: {\it If pp is a prime, n=n0+n1p+β‹―+nspsn=n_0+n_1p+\cdots +n_sp^s and m=m0+m1p+β‹―+mspsm=m_0+m_1p+\cdots +m_sp^s are the pp-adic expansions of nonnegative integers nn and mm, 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 uu-nomial coefficients, the Gaussian qq-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
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