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James Mc Laughlin

Publications and source records attributed to James Mc Laughlin.

At least 19 recordsLinked to original sources
arXiv2023

Some implications of Chu's $_{10}ψ_{10}$ extension of Bailey's $_{6}ψ_{6}$ summation formula

Lucy Slater used Bailey's $_6ψ_6$ 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 $_6ψ_6$ 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 $_6ψ_6$ summation formula and Jackson's $_6ϕ_5$ 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.

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arXiv2019

Some Observations on Lambert series, vanishing coefficients and dissections of infinite products and series

Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number of papers, that the coefficients in various arithmetic progressions in the series expansions of certain infinite $q$-products vanish. In the present paper it is shown that these results follow automatically (simply by specializing parameters) in an identity derived from a special case of Ramanujan's $_1ψ_1$ identity. Likewise, a number of authors have proven results about the $m$-dissections of certain infinite $q$-products using various methods. It is shown that many of these $m$-dissections also follow automatically (again simply by specializing parameters) from this same identity alluded to above. Two identities that mat be considered as extensions of two Identities of Ramanujan are also derived. It is also shown how applying similar ideas to certain other Lambert series gives rise to some rather curious $q$-series identities, such as, for any positive integer $m$, \begin{multline*} {\displaystyle \frac{\left(q,q,a,\frac{q}{a},\frac{b q}{d}, \frac{dq}{b}, \frac{aq}{b d}, \frac{b d q}{a};q\right)_{\infty }} {\left(b,\frac{q}{b},d,\frac{q}{d},\frac{a}{b},\frac{bq}{a},\frac{a}{d},\frac{dq}{a};q\right)_{\infty }}} = \sum _{r=0}^{m-1} q^r \frac{ \left(q^m,q^m,a q^{2 r},\frac{q^{m-2 r}}{a},\frac{b q^m}{d},\frac{d q^m}{b}, \frac{a q^m}{b d},\frac{b dq^m}{a};q^m\right){}_{\infty }} {\left(b q^r,\frac{q^{m-r}}{b},d q^r,\frac{q^{m-r}}{d},\frac{a q^r}{b},\frac{b q^{m-r}}{a},\frac{a q^r}{d}, \frac{dq^{m-r}}{a};q^m\right){}_{\infty }} \end{multline*} and \begin{equation*} (aq;q)_{\infty}\sum_{n=1}^{\infty} \frac{n a^n q^{n}}{(q;q)_n} = \sum_{r=1}^{m}(aq^{r};q^m)_{\infty}\sum_{n=1}^{\infty} \frac{na^n q^{n r}}{(q^m;q^m)_n}. \end{equation*} Applications to the Fine function $F(a,b;t)$ are also considered.

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arXiv2019

Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series

The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2ψ_2$ series \[ \sum_{n=-\infty}^{\infty}\frac{(a,c;q)_n}{(b,d;q)_n}z^n. \] Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. \\ New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say $χ(q)$, and a root of unity in a certain class, say $ζ$, that there is a theta function $θ_χ(q)$ such that \[ \lim_{q \to ζ}(χ(q) - θ_χ(q)) \] exists, as $q \to ζ$ from within the unit circle.

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arXiv2019

Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions

In this paper we show that various continued fractions for the quotient of general Ramanujan functions $G(aq,b,łq)/G(a,b,ł)$ may be derived from each other via Bauer-Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer-Muir transformations converge to the same limit. We also show that these continued fractions may be derived from Heine's continued fraction for a ratio of $_2ϕ_1$ functions and other continued fractions of a similar type, and by this method derive a new continued fraction for $G(aq,b,łq)/G(a,b,ł)$. Finally we derive a number of new versions of some beautiful continued fraction expansions of Ramanujan for certain combinations of infinite products, with the following being an example: \begin{multline*} \frac{(-a,b;q)_{\infty} - (a,-b;q)_{\infty}}{(-a,b;q)_{\infty}+ (a,-b;q)_{\infty}} = \frac{(a-b)}{1-a b} \- \frac{(1-a^2)(1-b^2)q}{1-a b q^2}\\ \- \frac{(a-bq^2)(b-aq^2)q}{1-a b q^4} %\phantom{sdsadadsaasdda}\\ \- \frac{(1-a^2q^2)(1-b^2q^2)q^3}{1-a b q^6} \- \frac{(a-bq^4)(b-aq^4)q^3}{1-a b q^8} \- \cds . \end{multline*}

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arXiv2019

Rogers-Ramanujan-Slater Type Identities

In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt to supply the earliest known reference for each identity. Also included are identities of false theta functions, along with their relationship to Rogers-Ramanujan type identities. We also describe several ways in which pairs/larger sets of identities may be related, as well as dependence relationships between identities.

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arXiv2019

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 $n \geq 1$, \[ p_{1,5}(m,5n)\geq p_{2,5}(m,5n), \,\,\,1 \leq m \leq 5n. \]

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arXiv2019

On a pair of identities from Ramanujan's lost notebook

Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive some general identities for integer partitions.

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arXiv2019

A Reciprocity Relation for WP-Bailey Pairs

We derive a new general transformation for WP-Bailey pairs by considering the a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.

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arXiv2019

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.

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arXiv2019

Further Results on Vanishing Coefficients in Infinite Product Expansions

We extend results of Andrews and Bressoud on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if \begin{equation*} \frac{(q^{r-tk}, q^{mk-(r-tk)}; q^{mk})_\infty}{(q^r,q^{mk-r}; q^{mk})_\infty} =: \sum_{n=0}^\infty c_nq^n, \end{equation*} 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, but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic results in terms of integer partitions.

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arXiv2019

Hybrid Proofs of the $q$-Binomial Theorem and other identities

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.

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arXiv2019

Some Applications of a Bailey-type Transformation

If $k$ is set equal to $a q$ in the definition of a WP Bailey pair, \[ β_{n}(a,k) = \sum_{j=0}^{n} \frac{(k/a)_{n-j}(k)_{n+j}}{(q)_{n-j}(aq)_{n+j}}α_{j}(a,k), \] this equation reduces to $β_{n}=\sum_{j=0}^{n}α_{j}$. This seemingly trivial relation connecting the $α_n$'s with the $β_n$'s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for poly-basic hypergeometric series.

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arXiv2019

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.

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arXiv2019

Continued Fraction Proofs of $m$-versions of Some Identities of Rogers-Ramanujan-Slater Type

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive \emph{$m$-versions} of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such $m$-versions of Rogers-Ramanujan-type identities.

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arXiv2019

A Hardy-Ramanujan-Rademacher-type formula for $(r,s)$-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 $p_{r,s}(n)$.

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arXiv2019

A New Summation Formula for WP-Bailey Pairs

Let $(α_n(a,k),β_n(a,k))$ be a WP-Bailey pair. Assuming the limits exist, let \[ (α_n^*(a),β_n^*(a))_{n\geq 1} = \lim_{k \to 1}\left(α_n(a,k),\frac{β_n(a,k)}{1-k}\right)_{n\geq 1} \] be the \emph{derived} WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive new basic hypergeometric summation and transformation formulae involving derived WP-Bailey pairs. We then use these formulae to derive new identities for various theta series/products which are expressible in terms of certain types of Lambert series.

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arXiv2019

Some new Transformations for Bailey pairs and WP-Bailey Pairs

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

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