246 research outputs found
On the asymptotic normality of the Legendre-Stirling numbers of the second kind
For the Legendre-Stirling numbers of the second kind asymptotic formulae are
derived in terms of a local central limit theorem. Thereby, supplements of the
recently published asymptotic analysis of the Chebyshev-Stirling numbers are
established. Moreover, we provide results on the asymptotic normality and
unimodality for modified Legendre-Stirling numbers
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
On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums
In this work we continue the investigation about the interplay between
hypergeometric functions and Fourier-Legendre () series
expansions. In the section "Hypergeometric series related to and
the lemniscate constant", through the FL-expansion of
(with ) we prove that all the hypergeometric
series
return rational
multiples of or the lemniscate constant, as
soon as is a polynomial fulfilling suitable symmetry constraints.
Additionally, by computing the FL-expansions of and
related functions, we show that in many cases the hypergeometric
function evaluated at can be
converted into a combination of Euler sums. In particular we perform an
explicit evaluation of In the
section "Twisted hypergeometric series" we show that the conversion of some
values into combinations of Euler sums,
driven by FL-expansions, applies equally well to some twisted hypergeometric
series, i.e. series of the form where is a
Stirling number of the first kind and
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