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James Mc Laughlin

Publications and source records attributed to James Mc Laughlin.

At least 37 records · Page 2Linked to original sources
arXiv2019

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 $j$, $1\leq j \leq 5$.

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arXiv2019

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\\ $(α_n(a,k)$, $β_n(a,k))$, in which $α_n(a,k)$ is independent of $k$, for generalizations of identities of the Rogers-Ramanujan type.

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arXiv2019

Lifting Bailey Pairs to WP-Bailey Pairs

A pair of sequences $(α_{n}(a,k,q),β_{n}(a,k,q))$ such that $α_0(a,k,q)=1$ and \[ β_{n}(a,k,q) = \sum_{j=0}^{n} \frac{(k/a; q)_{n-j}(k; q)_{n+j}}{(q;q)_{n-j}(aq;q)_{n+j}}α_{j}(a,k,q) \] is termed a \emph{WP-Bailey Pair}. Upon setting $k=0$ in such a pair we obtain a \emph{Bailey pair}. In the present paper we consider the problem of "lifting" a Bailey pair to a WP-Bailey pair, and use some of the new WP-Bailey pairs found in this way to derive some new identities between basic hypergeometric series and new single sum- and double sum identities of the Rogers-Ramanujan-Slater type.

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arXiv2019

Some new Families of Tasoevian- and Hurwitzian Continued Fractions

We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including $[0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1 }\,\,]_{n=0}^\infty$, $[0; \overline{c + d m^{n}}]_{n=1}^{\infty}$ and $[0; \overline{e u^{n}, f v^{ n}}]_{n=1}^\infty$. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurwitzian- and Tasoevian continued fractions of arbitrary long quasi-period, with arbitrarily many free parameters and whose limits can be determined as ratios of certain infinite series. We also derive expressions for arbitrarily long \emph{finite} continued fractions whose partial quotients lie in arithmetic progressions.

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arXiv2019

Some Observations on Khovanskii's Matrix Methods for extracting Roots of Polynomials

In this article we apply a formula for the $n$-th power of a $3\times 3$ matrix (found previously by the authors) to investigate a procedure of Khovanskii's for finding the cube root of a positive integer. We show, for each positive integer $α$, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to $α^{1/3}$. We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a more general class of cubic equations, and, for each such cubic, obtain a sequence of rationals that converge to the real root of the cubic. We prove that Khovanskii's method for finding the $m$-th ($m \geq 4$) root of a positive integer works, provided a free parameter is chosen to satisfy a very simple condition. Finally, we briefly consider another procedure of Khovanskii's, which also involves $m \times m$ matrices, for approximating the root of an arbitrary polynomial of degree $m$.

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arXiv2019

Continued Fractions and Generalizations with Many Limits: A Survey

There are infinite processes (matrix products, continued fractions, $(r,s)$-matrix continued fractions, recurrence sequences) which, under certain circumstances, do not converge but instead diverge in a very predictable way. We give a survey of results in this area, focusing on recent results of the authors.

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arXiv2019

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*}

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arXiv2019

A $q$-continued fraction

We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for $(q^2;q^3)_{\infty}/(q;q^3)_{\infty}$ and $(q;q^2)_\infty / (q^{3};q^{6})_\infty^3$. In addition, we give a new proof of the famous Rogers-Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.

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arXiv2019

Some more Long Continued Fractions, I

In this paper we show how to construct several infinite families of polynomials $D(\bar{x},k)$, such that $\sqrt{D(\bar{x},k)}$ has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter $k$. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.

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arXiv2019

Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields

For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where $C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$) and the fundamental solution can be written down. Likewise, if all the $X_{i}$'s and $t$ are non-negative then the continued fraction expansion of $\sqrt{F_{i}}$ can be written down. Furthermore, the congruence class modulo 4 of $F_{i}$ depends in a simple way on the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$ so that the fundamental unit can be written down for a large class of real quadratic fields. Along the way a complete solution is given to the problem of determining for which symmetric strings of positive integers $a_{1},..., a_{n}$ do there exist positive integers $D$ and $a_{0}$ such that $\sqrt{D} = [ a_{0};\bar{a_{1}, >..., a_{n},2a_{0}}]$.

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arXiv2018

Powers of a matrix and combinatorial identities

In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix. Finally, we use these results to derive various combinatorial identities.

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arXiv2018

On the Divergence in the General Sense of $q$-Continued Fraction on the Unit Circle

We show, for each $q$-continued fraction $G(q)$ in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which $G(q)$ diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other $q$-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.

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arXiv2018

Combinatorial Identities Deriving From The $N$-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.

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arXiv2018

The Convergence Behavior of $q$-Continued Fractions on the Unit Circle

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of $q$-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each $q$-continued fraction, $G(q)$, in this class, that there is an uncountable set of points, $Y_{G}$, on the unit circle such that if $y \in Y_{G}$ then $G(y)$ does not converge to a finite value. We discuss the implications of our theorems for the convergence of other $q$-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.

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arXiv2018

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 $\lim_{n \to \infty}|a_{n}| = \infty$.

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arXiv2018

Further Combinatorial Identities deriving from the $n$-th power of a $2 \times 2$ matrix

In this paper we use a formula for the $n$-th power of a $2\times2$ matrix $A$ (in terms of the entries in $A$) to derive various combinatorial identities. Three examples of our results follow. 1) We show that if $m$ and $n$ are positive integers and $s \in \{0,1,2,\dots,$ $\lfloor (mn-1)/2 \rfloor \}$, then \begin{multline*} \sum_{i,j,k,t}2^{1+2t-mn+n} \frac{(-1)^{nk+i(n+1)}}{1+δ_{(m-1)/2,\,i+k}} \binom{m-1-i}{i} \binom{m-1-2i}{k}\times\\ \binom{n(m-1-2(i+k))}{2j}\binom{j}{t-n(i+k)} \binom{n-1-s+t}{s-t}\\ =\binom{mn-1-s}{s}. \end{multline*} 2) The generalized Fibonacci polynomial $f_{m}(x,s)$ can be expressed as \[ f_{m}(x,s)= \sum_{k=0}^{\lfloor (m-1)/2 \rfloor}\binom{m-k-1}{k}x^{m-2k-1}s^{k}. \] We prove that the following functional equation holds: \begin{equation*} f_{mn}(x,s)=f_{m}(x,s)\times f_{n}\left (\,f_{m+1}(x,s)+sf_{m-1}(x,s), \,-(-s)^{m}\right) . \end{equation*} 3) If an arithmetical function $f$ is multiplicative and for each prime $p$ there is a complex number $g(p)$ such that \begin{equation*} f(p^{n+1}) = f(p)f(p^{n})- g(p)f(p^{n-1}), \hspace{15pt} n \geq 1, \end{equation*} then $f$ is said to be \emph{specially multiplicative}. We give another derivation of the following formula for a specially multiplicative function $f$ evaluated at a prime power: \begin{equation*} f(p^{k})=\sum_{j=0}^{\lfloor k/2 \rfloor}(-1)^{j} \binom{k-j}{j}f(p)^{k-2j}g(p)^{j}. \end{equation*} We also prove various other combinatorial identities.

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