546 research outputs found
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by 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 coefficients onto
coefficients interpretations thus bringing us back to
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
Zeckendorf's theorem states that every positive integer can be written
uniquely as a sum of non-consecutive Fibonacci numbers , with initial
terms . We consider the distribution of the number of
summands involved in such decompositions. Previous work proved that as the distribution of the number of summands in the Zeckendorf
decompositions of , appropriately normalized, converges
to the standard normal. The proofs crucially used the fact that all integers in
share the same potential summands.
We generalize these results to subintervals of as ; the analysis is significantly more involved here as different integers
have different sets of potential summands. Explicitly, fix an integer sequence
. As , for almost all the distribution of the number of summands in the Zeckendorf
decompositions of integers in the subintervals ,
appropriately normalized, converges to the standard normal. The proof follows
by showing that, with probability tending to , has at least one
appropriately located large gap between indices in its decomposition. We then
use a correspondence between this interval and 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)
In 1878 \'E. Lucas proved a remarkable result which provides a simple way to
compute the binomial coefficient modulo a prime in terms of
the binomial coefficients of the base- digits of and : {\it If is
a prime, and are the
-adic expansions of nonnegative integers and , 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 -nomial
coefficients, the Gaussian -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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