Combinatorial Identities Deriving From The NN-th Power Of A 2\Times 2 Matrix

In this paper we give a new formula for the $n$-th power of a $2\times2$ matrix. More precisely, we prove the following: Let $A= \left ( \begin{matrix} a & b \\ c & d \end{matrix} \right )$ be an arbitrary $2\times2$ matrix, $T=a+d$ its trace, $D= ad-bc$ its determinant and define $$y_{n} :\,= \sum_{i=0}^{\lfloor n/2 \rfloor}\binom{n-i}{i}T^{n-2 i}(-D)^{i}.$$ Then, for $n \geq 1$, \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 $n$-th power of a matrix, various matrix identities, formulae for the $n$-th power of particular matrices, etc, to derive various combinatorial identities.Comment: 13 page

An Identity Motivated by an Amazing Identity of Ramanujan

Ramanujan stated an identity to the effect that if three sequences $\{a_n\}$, $\{b_n\}$ and $\{c_n\}$ are defined by $r_1(x)=:\sum_{n=0}^{\infty}a_nx^n$, $r_2(x)=:\sum_{n=0}^{\infty}b_nx^n$ and $r_3(x)=:\sum_{n=0}^{\infty}c_nx^n$ (here each $r_i(x)$ is a certain rational function in $x$), then $$a_n^3+b_n^3-c_n^3=(-1)^n, \hspace{25pt} \forall \,n \geq 0.$$ 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 $j$, $1\leq j \leq 5$.Comment: 5 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 $M\geq 5$ is an integer and the integers $a$ and $b$ are relatively prime to $M$ and satisfy $1\leq a<b<M/2$, and the $c(m,n)$ are defined by $$\frac{1}{(sq^a,sq^{M-a};q^M)_{\infty}}-\frac{1}{(sq^b,sq^{M-b};q^M)_{\infty}}:=\sum_{m,n\geq 0} c(m,n)s^m q^n,$$ then $c(m, Mn)\geq 0$ for all integers $m\geq 0, n\geq 0$. %If, in addition, $M$ is even, then $c(m, Mn+M/2)\geq 0$ for all integers $m\geq 0, n\geq 0$. A similar result is proved for the integers $d(m,n)$ defined by $$(-sq^a,-sq^{M-a};q^M)_{\infty}-(-sq^b,-sq^{M-b};q^M)_{\infty}:=\sum_{m,n\geq 0} d(m,n)s^m q^n.$$ In each case there are obvious interpretations in terms of integer partitions. For example, if $p_{1,5}(m,n)$ (respectively $p_{2,5}(m,n)$) denotes the number of partitions of $n$ into exactly $m$ parts $\equiv \pm 1 (\mod 5)$ (respectively $\equiv \pm 2 (\mod 5)$), then for each integer $n \geq 1$, $$p_{1,5}(m,5n)\geq p_{2,5}(m,5n), \,\,\,1 \leq m \leq 5n.$$Comment: 11 page

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 $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some $q$ orthogonal polynomials, and various multi-sums that are expressible as infinite products.Comment: 20 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\\ $(\alpha_n(a,k)$, $\beta_n(a,k))$, in which $\alpha_n(a,k)$ is independent of $k$, for generalizations of identities of the Rogers-Ramanujan type.Comment: 17 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 (r,s)(r,s)-regular partitions

Let $p_{r,s}(n)$ denote the number of partitions of a positive integer $n$ into parts containing no multiples of $r$ or $s$, where $r>1$ and $s>1$ are square-free, relatively prime integers. We use classical methods to derive a Hardy-Ramanujan-Rademacher-type infinite series for $p_{r,s}(n)$.Comment: 19 page

Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field

Let $p \geq 5$ be a prime and for $a, b \in \mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\mathbb{F}_{p}$ with equation $y^2=x^3+a\,x + b$. As usual define the trace of Frobenius $a_{p,\,a,\,b}$ 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 $\sum_{t\in\mathbb{F}_{p}} a_{p,\, t,\, b}$, $\sum _{t \in \mathbb{F}_{p}} a_{p,\,a,\, t}$, $\sum_{t=0}^{p-1}a_{p,\,t,\,b}^{2}$, $\sum_{t=0}^{p-1}a_{p,\,a,\,t}^{2}$ and $\sum_{t=0}^{p-1}a_{p,\,t,\,b}^{3}$ for primes $p$ in various congruence classes. As an example of our results, we prove the following: Let $p \equiv 5$ $($mod 6$)$ be prime and let $b \in \mathbb{F}_{p}^{*}$. 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

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 $q$ continued fraction to show, that if $G(q)$ is one of these continued fractions and $|q|>1$, then either $G(q)$ converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction $K_{n=1}^{\infty}a_{n}/1$ converge to different values, then $\lim_{n \to \infty}|a_{n}| = \infty$.Comment: 11 page

Polynomial Continued Fractions

Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than or equal to one. Here we study cases of higher degree for both numerator and denominator polynomials, with particular attention given to cases in which the degrees are equal. We extend work of Ramanujan on continued fractions with rational limits and also consider cases where the limits are irrational.Comment: 13 page