860 research outputs found
An Identity Motivated by an Amazing Identity of Ramanujan
Ramanujan stated an identity to the effect that if three sequences ,
and are defined by ,
and
(here each is a certain rational function in ), then Motivated by
this amazing identity, we state and prove a more general identity involving
eleven sequences, the new identity being "more general" in the sense that
equality holds not just for the power 3 (as in Ramanujan's identity), but for
each power , .Comment: 5 page
Combinatorial Identities Deriving From The -th Power Of A 2\Times 2 Matrix
In this paper we give a new formula for the -th power of a
matrix.
More precisely, we prove the following: Let be an arbitrary matrix, its
trace,
its determinant and define Then, for ,
\begin{equation*} A^{n}=\left ( \begin{matrix} y_{n}-d \,y_{n-1} & b \,y_{n-1}
\\ c\, y_{n-1}& y_{n}-a\, y_{n-1} \end{matrix} \right ). \end{equation*}
We use this formula together with an existing formula for the -th power of
a matrix, various matrix identities, formulae for the -th power of
particular matrices, etc, to derive various combinatorial identities.Comment: 13 page
Small Prime Powers in the Fibonacci Sequence
It is shown that there are no non-trivial fifth-, seventh-, eleventh-,
thirteenth- or seventeenth powers in the Fibonacci sequence. For eleventh,
thirteenth- and seventeenth powers an alternative (to the usual exhaustive
check of products of powers of fundamental units) method is used to overcome
the problem of having a large number of independent units and relatively high
bounds on their exponents. It is envisaged that the same method can be used to
decide the question of the existence of higher small prime powers in the
Fibonacci sequence and that the method can be applied to other binary
recurrence sequences. The alternative method mentioned may have wider
applications.Comment: 22 pages. In the previous version of this paper my use of the GP/PARI
command "bnfinit" meant that further work was necessary to show that the
systems of units produced were indeed fundamental systems. The necessary
further work has been included here (see page 10 of the paper for a more
detailed explanation
General Multi-sum Transformations and Some Implications
We give two general transformations that allows certain quite general basic
hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary
sequence ), to be reduced to an infinite -product times a single
basic hypergeometric sum. Various applications are given, including summation
formulae for some orthogonal polynomials, and various multi-sums that are
expressible as infinite products.Comment: 20 page
Refinements of Some Partition Inequalities
In the present paper we initiate the study of a certain kind of partition
inequality, by showing, for example, that if is an integer and the
integers and are relatively prime to and satisfy ,
and the are defined by then for all integers .
%If, in addition, is even, then for all integers
. A similar result is proved for the integers
defined by In each case there are obvious interpretations in terms of
integer partitions. For example, if (respectively
) denotes the number of partitions of into exactly parts
(respectively ), then for each
integer , Comment: 11 page
General WP-Bailey Chains
Motivated by a recent paper of Liu and Ma, we describe a number of general
WP-Bailey chains. We show that many of the existing WP-Bailey chains (or
branches of the WP-Bailey tree), including chains found by Andrews, Warnaar and
Liu and Ma, arise as special cases of these general WP-Bailey chains. We
exhibit three new branches of the WP-Bailey tree, branches which also follow as
special cases of these general WP-Bailey chains. Finally, we describe a number
of new transformation formulae for basic hypergeometric series which arise as
consequences of these new WP-Bailey chains.Comment: 20 page
A Hardy-Ramanujan-Rademacher-type formula for -regular partitions
Let denote the number of partitions of a positive integer
into parts containing no multiples of or , where and are
square-free, relatively prime integers. We use classical methods to derive a
Hardy-Ramanujan-Rademacher-type infinite series for .Comment: 19 page
Some Implications of the WP-Bailey Tree
We consider a special case of a WP-Bailey chain of George Andrews, and use it
to derive a number of curious transformations of basic hypergeometric series.
We also derive two new WP-Bailey pairs, and use them to derive some additional
new transformations for basic hypergeometric series. Finally, we briefly
consider the implications of WP-Bailey pairs\\ ,
, in which is independent of , for
generalizations of identities of the Rogers-Ramanujan type.Comment: 17 page
A Theorem on Divergence in the General Sense for Continued Fractions
If the odd and even parts of a continued fraction converge to different
values, the continued fraction may or may not converge in the general sense. We
prove a theorem which settles the question of general convergence for a wide
class of such continued fractions. We apply this theorem to two general classes
of continued fraction to show, that if is one of these continued
fractions and , then either converges or does not converge in the
general sense. We also show that if the odd and even parts of the continued
fraction converge to different values, then .Comment: 11 page
Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field
Let be a prime and for , let
denote the elliptic curve over with equation . As usual define the trace of Frobenius by \begin{equation*}
\#E_{a,b}(\mathbb{F}_{p}) = p+1 -a_{p,\,a,\,b}. \end{equation*} We use
elementary facts about exponential sums and known results about binary
quadratic forms over finite fields to evaluate the sums
, ,
,
and for primes in various congruence
classes.
As an example of our results, we prove the following: Let mod
6 be prime and let . Then \begin{equation*}
\sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}= -p\left((p-2)\left(\frac{-2}{p}\right)
+2p\right)\left(\frac{b}{p}\right). \end{equation*}Comment: 16 page
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