221 research outputs found
New harmonic number identities with applications
We determine the explicit formulas for the sum of products of homogeneous
multiple harmonic sums when
. We apply these identities to the study of two
congruences modulo a power of a prime
The dinner table problem: the rectangular case
people are seated randomly at a rectangular table with and seats along the two opposite sides for two
dinners. What's the probability that neighbors at the first dinner are no more
neighbors at the second one? We give an explicit formula and we show that its
asymptotic behavior as goes to infinity is (it is known
that it is for a round table). A more general permutation
problem is also considered.Comment: 10 page
Supercongruences for a truncated hypergeometric series
The purpose of this note is to obtain some congruences modulo a power of a
prime involving the truncated hypergeometric series for and . In the last
section, the special case is considered.Comment: Revised versio
Supercongruences for the Almkvist-Zudilin numbers
Given a prime number , the study of divisibility properties of a sequence
has two contending approaches: -adic valuations and
superconcongruences. The former searches for the highest power of dividing
, for each ; while the latter (essentially) focuses on the maximal
powers and such that is congruent to modulo
. This is called supercongruence. In this paper, we prove a conjecture on
supercongruences for sequences that have come to be known as the
Almkvist-Zudilin numbers. Some other (naturally) related family of sequences
will be considered in a similar vain.Comment: 15 page
Congruences for central binomial sums and finite polylogarithms
We prove congruences, modulo a power of a prime p, for certain finite sums
involving central binomial coefficients
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