Papers by James Mc Laughlin
arXiv (Cornell University), Jan 5, 2019
Lucy Slater used Bailey's 6 ψ 6 summation formula to derive the Bailey pairs she used to construc... more Lucy Slater used Bailey's 6 ψ 6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's 10 ψ 10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs. In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new general Bailey pairs (containing one or more free parameters) which are also implied by the 6 ψ 6 summation formula. Slater used the Jacobi triple product identity (sometimes coupled with the quintuple product identity) to derive her infinite products. Here we also use other summation formulae (including special cases of the 6 ψ 6 summation formula and Jackson's 6 φ 5 summation formula) to derive some of our infinite products. We use the new Bailey pairs, and/or the summation methods mentioned above, to give new proofs of some general series-product identities due to Ramanujan, Andrews and others. We also derive a new general series-product identity, one which may be regarded as a partner to one of the Ramanujan identities. We also find new transformation formulae between basic hypergeometric series, new identities of Rogers-Ramanujan type, and new false theta series identities. Some of these latter are a kind of "hybrid" in that one side of the identity consists a basic hypergeometric series, while the other side is formed from a theta product multiplied by a false theta series. This type of identity appears to be new.
Electronic Journal of Combinatorics , 2011
We give “hybrid” proofs of the q-binomial theorem and other identities. The proofs are “hybrid” i... more We give “hybrid” proofs of the q-binomial theorem and other identities. The proofs are “hybrid” in the sense that we use partition arguments to prove
a restricted version of the theorem, and then use analytic methods (in the form of
the Identity Theorem) to prove the full version.
We prove three somewhat unusual summation formulae, and use these to give
hybrid proofs of a number of identities due to Ramanujan.
Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.
We give hybrid proofs of the q-binomial theorem and other identities. The proofs are hybrid in th... more We give hybrid proofs of the q-binomial theorem and other identities. The proofs are hybrid in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities
Bulletin of the Australian Mathematical Society, 2007
Let p ≥ 5 be a prime and for a, b ε p, let Ea, b denote the elliptic curve over p with equation... more Let p ≥ 5 be a prime and for a, b ε p, let Ea, b denote the elliptic curve over p with equation y2 = x3 + ax + b. As usual define the trace of Frobenius ap, a, b by We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums and for primes p in various congruence classes.As an example of our results, we prove the following: Let p ≡ 5 (mod 6) be prime and let b ε *p. Then
International Mathematical Forum, 2010
Abstract. If k is set equal to aq in the definition of a WP Bailey pair, ... (k/a)n−j (k)n+j (q)n... more Abstract. If k is set equal to aq in the definition of a WP Bailey pair, ... (k/a)n−j (k)n+j (q)n−j (aq)n+j αj (a, k), ... We begin by recalling a construction of Andrews [1]. If a pair of sequences (αn(a, k), βn(a, k)) satisfy ... (k/a)n−j(k)n+j (q)n−j(aq)n+j αj(a, k), ... (1 − cq2j)(ρ1,ρ2)j(k/c)n ...
We prove a generalization of Schroter\u27s formula to a product of an arbitrary number of Jacobi ... more We prove a generalization of Schroter\u27s formula to a product of an arbitrary number of Jacobi triple products. It is then shown that many of the well-known identities involving Jacobi triple products (for example the Quintuple Product Identity, the Septuple Product Identity, and Winquist\u27s Identity) all then follow as special cases of this general identity. Various other general identities, for example certain expansions of (q; q)(infinity) and (q; q)(infinity)(k), k \u3e= 3, as combinations of Jacobi triple products, are also proved
Finding polynomial solutions to Pell’s equation is of interest as such solutions sometimes allow ... more Finding polynomial solutions to Pell’s equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields. In this paper, for each triple of positive integers (c, h, f) satisfying c 2 − f h2 = 1, where (c, h) are the smallest pair of integers satisfying this equation, several sets of polynomials (c(t), h(t), f(t)) which satisfy c(t) 2 − f(t) h(t) 2 = 1 and (c(0), h(0), f(0)) = (c, h, f) are derived. Moreover, it is shown that the pair (c(t), h(t)) constitute the fundamental polynomial solution to the Pell’s equation above. The continued fraction expansion of p f(t) is given in certain general cases (for example, when the continued fraction expansion of √ f has odd period length, or even period length or has period length ≡ 2 mod 4 and the middle quotient has a particular form etc). Some applications to determining the fundamental unit in real quadratic fields is also discussed
Let a, b, c, d be complex numbers with d 6= 0 and |q| \u3c 1. Define H1(a, b, c, d, q) := 1 1 + −... more Let a, b, c, d be complex numbers with d 6= 0 and |q| \u3c 1. Define H1(a, b, c, d, q) := 1 1 + −abq + c (a + b)q + d + · · · + −abq2n+1 + cqn (a + b)q n+1 + d + · · · . We show that H1(a, b, c, d, q) converges and 1 H1(a, b, c, d, q) − 1 = c − abq d + aq P∞ j=0 (b/d) j (−c/bd)j q j(j+3)/2 (q)j (−aq2/d)j P∞ j=0 (b/d) j (−c/bd)j q j(j+1)/2 (q)j (−aq/d)j . We then use this result to deduce various corollaries, including the following: 1 1 − q 1 + q − q 3 1 + q 2 − q 5 1 + q 3 − · · · − q 2n−1 1 + q n − · · · = (q 2 ; q 3 )∞ (q; q 3)∞ , (−aq)∞ X∞ j=0 (bq) j (−c/b)j q j(j−1)/2 (q)j (−aq)j = (−bq)∞ X∞ j=0 (aq) j (−c/a)j q j(j−1)/2 (q)j (−bq)j , and the Rogers-Ramanujan identities, X∞ n=0 q n 2 (q; q)n = 1 (q; q 5)∞(q 4; q 5)∞ , X∞ n=0 q n 2+n (q; q)n = 1 (q 2; q 5)∞(q 3; q 5)∞
Solving Pell’s equation is of relevance in finding fundamental units in real quadratic fields and... more Solving Pell’s equation is of relevance in finding fundamental units in real quadratic fields and for this reason polynomial solutions are of interest in that they can supply the fundamental units in infinite families of such fields. In this paper an algorithm is described which allows one to construct, for each positive integer n, a finite collection, {Fi}, of multi-variable polynomials (with integral coefficients), each satisfying a multi-variable polynomial Pell’s equation C 2 i − FiH 2 i = (−1)n−1 , where Ci and Hi are multi-variable polynomials with integral coefficients. Each positive integer whose square-root has a regular continued fraction expansion with period n + 1 lies in the range of one of these polynomials. Moreover, the continued fraction expansion of these polynomials is given explicitly as is the fundamental solution to the above multi-variable polynomial Pell’s equation. Some implications for determining the fundamental unit in a wide class of real quadratic field...
Lucy Slater used Bailey\u27s 6ψ6 summation formula to derive the Bailey pairs she used to constru... more Lucy Slater used Bailey\u27s 6ψ6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu\u27s 10ψ10 generalization of Bailey\u27s formula to produce quite general Bailey pairs. Slater\u27s Bailey pairs are then recovered as special limiting cases of these more general pairs. In re-examining Slater\u27s work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new general Bailey pairs (containing one or more free parameters) which are also implied by the 6ψ6 summation formula. Slater used the Jacobi triple product identity (sometimes coupled with the quintuple product identity) to derive her infinite products. Here we also use other summation formulae (including special cases of the 6ψ6 summation formula and Jackson\u27s 6Ø5 summation formula) to de...
Integers, 2007
In this article we apply a formula for the n-th power of a 3×3 matrix (found previously by the au... more In this article we apply a formula for the n-th power of a 3×3 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 ≥ 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 × m matrices, for approximating the root of an arbitrary polynomial of degree m.
INTEGERS: The Electronic Journal of Combinatorial Number Theory, 2005
In this article we obtain a general polynomial identity in k variables, where k ≥ 2 is an arbitra... more In this article we obtain a general polynomial identity in k variables, where k ≥ 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×k matrix. Finally, we use these results to derive various combinatorial identities.
The Ramanujan Journal, 2022
For integers t and m with m ≥ 5 relatively prime to 6 such that 1 ≤ t < m/2 and gcd(t, m) = 1, de... more For integers t and m with m ≥ 5 relatively prime to 6 such that 1 ≤ t < m/2 and gcd(t, m) = 1, define Q(t, m) := (q 2t , q m−2t , q m ; q m)∞ (q t , q m−t ; q m)∞ .
The Electronic Journal of Combinatorics, 2011
We provide finite analogs of a pair of two-variable $q$-series identities from Ramanujan's lo... more We provide finite analogs of a pair of two-variable $q$-series identities from Ramanujan's lost notebook and a companion identity.
Journal of Mathematical Analysis and Applications, 2017
Lucy Slater used Bailey's 6ψ6 summation formula to derive the Bailey pairs she used to construct ... more Lucy Slater used Bailey's 6ψ6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's 10ψ10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs. In reexamining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new general Bailey pairs (containing one or more free parameters) which are also implied by the 6ψ6 summation formula. Slater used the Jacobi triple product identity (sometimes coupled with the quintuple product identity) to derive her infinite products. Here we also use other summation formulae (including special cases of the 6ψ6 summation formula and Jackson's 6φ5 summation formula) to derive some of our infinite products. We use the new Bailey pairs, and/or the summation methods mentioned above, to give new proofs of some general series-product identities due to Ramanujan, Andrews and others. We also derive a new general series-product identity, one which may be regarded as a partner to one of the Ramanujan identities. We also find new transformation formulae between basic hypergeometric series, new identities of Rogers-Ramanujan type, and new false theta series identities. Some of these latter are a kind of "hybrid" in that one side of the identity consists a basic hypergeometric series, while the other side is formed from a theta product multiplied by a false theta series. This type of identity appears to be new.
Combinatorial Number Theory, 2009
We provide the missing member of a family of four q-series identities related to the modulus 36, ... more We provide the missing member of a family of four q-series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combinatorial implications of the identities in this family, and of some of the identities we considered in "Identities of the Ramanujan-Slater type related to the moduli 18 and 24," [J. Math.
The Ramanujan Journal, 2015
We give two general transformations that allows certain quite general basic hypergeometric multi-... more 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 multisums that are expressible as infinite products.
Journal of the Australian Mathematical Society, 2014
We extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions... more We extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if $$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$ for certain integers $k$, $m$, $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon [‘Vanishing coefficients in the expansion of products of Rogers–Ramanujan type’, in: The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemporary Mathematics, 166 (American Mathematical Society, Providence, RI, 1994), 129–139], but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic...
The Ramanujan Journal, 2012
Let p r,s (n) denote the number of partitions of a positive integer n into parts containing no mu... more 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). Keywords q-series • partitions • circle-method • Hardy-Ramanujan-Rademacher Mathematics Subject Classification (2000) Primary 11P82 • Secondary 05A17 • 11L05 • 11D85 • 11P55 • 11Y35 1 Introduction A partition of a positive integer n is a representation of n as a sum of positive integers, where the order of the summands does not matter. We use p(n) to denote the number of partitions of n, so that, for example, p(4) = 5, since 4 may be represented as 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1. The function p(n) increases rapidly with n, and it is difficult to compute p(n) directly for large n. Rademacher [23], by slightly modifying earlier work of Hardy and Ramanujan [13], derived a remarkable infinite series for p(n). To describe this series
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Papers by James Mc Laughlin
a restricted version of the theorem, and then use analytic methods (in the form of
the Identity Theorem) to prove the full version.
We prove three somewhat unusual summation formulae, and use these to give
hybrid proofs of a number of identities due to Ramanujan.
Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.
a restricted version of the theorem, and then use analytic methods (in the form of
the Identity Theorem) to prove the full version.
We prove three somewhat unusual summation formulae, and use these to give
hybrid proofs of a number of identities due to Ramanujan.
Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities.