James Mc Laughlin

West Chester University · Department of Mathematics

In the present paper, we consider some variations and generalizations of the multi-sum to single-sum transformation recently used by Rosengren in his proof of the Kanade–Russell identities. These general transformations are then used to prove a number of identities equating multi-sums and infinite products or multi-sums and infinite product \(\time...

This paper examines the phenomenon of coefficients that vanish in a class of infinite products related to those first examined by Hirschhorn, and later by Tang, Baruah and Kaur, and the first author of this paper. Several infinite families of vanishing coefficient phenomena are found. In particular, it is shown that if [Formula: see text] is a prim...

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Motivated by results of Hirschhorn, Tang, and Baruah and Kaur on vanishing coefficients (in arithmetic progressions) in a new class of infinite product which have appeared recently, we further examine such infinite products, and find that many such results on vanishing coefficients may be grouped into families. For example, one result proved in the...

We prove a generalization of Schröter’s 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’s Identity) all then follow as special cases of this general...

We prove a generalization of Schröter’s 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’s Identity) all then follow as special cases of this general...

In this paper we show that various continued fractions for the quotient of general Ramanujan functions $G(aq,b,\l q)/G(a,b,\l)$ 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 t...

The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2\psi_2$ 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 formula...

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...

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...

We provide finite analogs of a pair of two-variable $q$-series identities from Ramanujan's lost notebook and a companion identity.

Lucy Slater used Bailey's $_6\psi_6$ 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 $_{10}\psi_{10}$ generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then r...

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...

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...

If $k$ is set equal to $a q$ in the definition of a WP Bailey pair, \[ \beta_{n}(a,k) = \sum_{j=0}^{n} \frac{(k/a)_{n-j}(k)_{n+j}}{(q)_{n-j}(aq)_{n+j}}\alpha_{j}(a,k), \] this equation reduces to $\beta_{n}=\sum_{j=0}^{n}\alpha_{j}$. This seemingly trivial relation connecting the $\alpha_n$'s with the $\beta_n$'s has some interesting consequences,...

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...

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 $\alpha$, how to construct certain families of integer sequences such that a certain rational expression, inv...

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.

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...

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.

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 the...

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 2...

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 par...

In this paper we show that various continued fractions for the quotient of general Ramanujan functions G(aq,b,λq)/G(a,b,λ) 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...

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 polynomial...

We extend results of Andrews and Bressoud ['Vanishing coefficients in infinite product expansions', J. Aust. Math. Soc. Ser. A 27(2) (1979), 199-202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if [GRAPHICS] for certain integers k, m, s and t, where r = sm + t, then c(kn r...

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. Another consequence is a rather unusual summation formul...

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 s...

We derive a new general transformation for WP-Bailey pairs by considering 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 iden...

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).

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))(infinity)/(q;q(3))(infinity) and (q;q(2))(infinity)/(...

We provide finite analogs of a pair of two-variable q-series identi- ties from Ramanujan's lost notebook and a companion identity.

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,...

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≥1 =lim k→1 α n (a,k),β n (a,k) 1-k n≥1 be the 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...

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. KeywordsBailey pairs-WP-Bailey Chains-WP-Bailey...

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,...

Ramanujan stated an identity to the effect that if three sequences {a n},{b n} and {c n} are defined by r1(x) =: Σ ∞n=0a nx n,r2(x) =: σ ∞n=0b nx n and r3(x) =: σ ∞n=0c nx n (here each ri(x) is a certain rational function in a;), then a 3n + b 3n-c 3n = for all n ≥ 0. Motivated by this amazing identity, we state and prove a more general identity in...

If k is set equal to aq in the definition of a WP Bailey pair, β n (a, k) = n X j=0 (k/a)n−j(k)n+j (q) n−j (aq) n+j α j (a, k), this equation reduces to βn = P n j=0 αj. 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...

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...

Lucy Slater used Bailey’s 6 ψ 6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 Rogers-Ramanujan type identities. 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 l...

We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers-Ramanujan type and identities of false theta functions.

A pair of sequences (α n (a,k,q),β(a,k,q)) such that α 0 (a,k,q)=1 and β n (a,k,q)=∑ j=0 n (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 WP-Bailey pair. Upon setting k=0 in such a pair we obtain a 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...

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 pai...

We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3-and 4-term transformation formulae between basic hypergeometric series.

We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function ide...

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...

We reformulate two construction of George Andrews (con-structions which enable new WP-Bailey pairs to be derived from ex-isting WP-Bailey pairs) as transformations between WP-Bailey Pairs (α n (a, k), β n (a, k)). We then show how a special case of one of these transformations can be used to immediately derive a finitization of any series-product i...

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 combina-torial 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...

We derive closed-form expressions for several new classes of Hurwitzian-and Tasoevian continued fractions, including [0; p − 1, 1, u(a + 2nb) − 1, p − 1, 1, v(a + (2n + 1)b) − 1 ] ∞ n=0 , [0; c + dm n ] ∞ n=1 and [0; eu n , f v n ] ∞ n=1. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurw...

Given a sequence of complex square matrices, $a_n$, consider the sequence of their partial products, defined by $p_n=p_{n-1}a_{n}$. What can be said about the asymptotics as $n\to\infty$ of the sequence $f(p_n)$, where $f$ is a continuous function? A special case of our most general result addresses this question under the assumption that the matri...

For integers m⩾2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem.We give a theorem on a class of Poincaré-type recurrences...

In this paper we show how to construct several infinite fam-ilies of polynomials D(¯ x, k), such that p D(¯ 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...

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 , how to construct certain families of integer sequences such that a certain rational expression, involving the ratio o...

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...

In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods.

We consider two classes of q-continued fraction whose odd and even parts are limit 1-periodic for |q| > 1, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.

In this paper we use a formula for the nth power of a 2 × 2 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 ∈ { 0, 1, 2,..., ⌊ ( mn - 1 ) / 2 ⌋ }, then{A formula is presented}. (2) The generalized Fibonacci polynomial f...

In this paper we present a convergence theorem for continued fractions of the form . By deriving conditions on the an which ensure that the odd and even parts of converge, these same conditions also ensure that they converge to the same limit. Examples will be given.

In this article we obtain a general polynomial identity in k variables, where k ≥ 2i s 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.

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...

This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t∈(0,1) be denoted by [0,e 1 (t),e 2 (t),...] and let the i-th convergent of this continued fraction expansion be denoted by c i (t)/d i (t). Let S={t∈(0,1):e i+1 (t)≥ϕ d i (t) infinitelyoften...

Let $f(x) \in \mathbb{Z}[x]$. Set $f_{0}(x) = x$ and, for $n \geq 1$, define $f_{n}(x)$ $=$ $f(f_{n-1}(x))$. We describe several infinite families of polynomials for which the infinite product \prod_{n=0}^{\infty} (1 + \frac{1}{f_{n}(x)}) has a \emph{specializable} continued fraction expansion of the form S_{\infty} = [1;a_{1}(x), a_{2}(x), a_{3}(x...

In this paper we show how to apply various techniques and theorems (including Pincherle's theorem, an extension of Euler's formula equating infinite series and continued fractions, an extension of the corresponding transformation that equates infinite products and continued fractions, extensions and contractions of continued fractions and the Bauer...

If a continued fraction $K_{n=1}^{\infty} a_{n}/b_{n}$ is known to converge but its limit is not easy to determine, it may be easier to use an extension of $K_{n=1}^{\infty}a_{n}/b_{n}$ to find the limit. By an extension of $K_{n=1}^{\infty} a_{n}/b_{n}$ we mean a continued fraction $K_{n=1}^{\infty} c_{n}/d_{n}$ whose odd or even part is $K_{n=1}^...

In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m \geq 1$ are integers. The limits of such continued fractions, for general $a$ and in the cases $m=1$ and $m=2$...

Solving Pell's equation is of relevance in finding fundamen-tal units in real quadratic fields and for this reason polynomial solutions are of interest in that they can supply the fundamen-tal 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 colle...

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...

Let the continued fraction expansion of any irrational number $t \in (0,1)$ be denoted by $[0,a_{1}(t),a_{2}(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \[ S=\{t \in (0,1): a_{i+1}(t) \geq \phi^{d_{i}(t)} \text{infinitely often}\}, \] where $\phi = (\sqrt{5}+1)/2$. Let $Y_{S} =\{\...

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...

In the present paper we examine a number of variations of the Bailey transform and use these to derive several transformations of basic hypergeometric series which we believe to be new. One quite interesting discovery, which we also believe to be new, is a link between the Prouhet-Tarry-Escott problem and one of these transformations. We also use s...

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...

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

In this paper we give a new formula for the n-th power of a 2 ◊ 2 matrix. More precisely, we prove the following: Let A = a b c d be an arbitrary 2◊2 matrix,

Let (fin(a;k);fln(a;k)) be a WP-Bailey pair. Assuming the limits exist, let (fi⁄ n(a);fl ⁄ n(a)) = lim k!1 fin(a;k); fln(a;k) 1 ¡ k be the derived WP-Bailey pair. By considering a particular limiting case of a transformation due to George Andrews, we derive some trans- formation and summation formulae for derived WP-Bailey pairs. We then use the fo...

Let p 5 be a prime and for a,b 2 Fp, let Ea,b denote the elliptic curve over Fp with equation y2 = x3 + ax + b. As usual define the trace of Frobenius ap, a, b by #Ea,b(Fp) = p + 1 ap, a, b. We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums P t2Fp ap, t, b, P t2Fp a...

We examine certain limiting cases of a WP-Bailey chain discovered by George Andrews, and of certain classical summation for- mulae for basic hypergeometric series. From these we derive new trans- formations connecting some basic hypergeometric series to certain sums of Lambert series. We then use known expressions for q-products expressible in term...