On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews’ partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order f(q), ?(q) and ψ(q). An identity of Ramanujan is proved combinatorially. Several new identities are also established.     

An Expansion Formula for q-Series and Applications

In this paper the author proves a q-expansion formula which utilizes the Leibniz formula for the q-differential operator. This expansion leads to new proofs of the Rogers–Fine identity, the nonterminating ₆₅ summation formula, and Watson's q-analog of Whipple's theorem. Andrews' identities for sums of three squares and sums of three triangular numbers are also derived. Other identities of Andrews and new identities for Hecke type series are also discussed.     

Circle and divisor problems,and double series of Bessel functions

In approximately 1915, Ramanujan recorded two identities involving doubly infinite series of Bessel functions. The identities were brought to the mathematical public for the first time when his lost notebook was published in 1988, and are connected with the classical, long-standing circle and divisor problems, respectively. We provide a proof of the first identity for the first time by analytically continuing a new kind of Dirichlet series. Delicate estimates of exponential sums are needed, and the new methods we introduce may be of independent interest.     

Com plete p-elli ptic integrals and a com putation formula of $$\ pi _ p$$π p for $$ p=4$$ p=4

We study the q-bracket operator of Bloch and Okounkov, recently examined by Zagier and other authors, when applied to functions defined by two classes of sums over the parts of an integer partition. We derive convolution identities for these functions and link both classes of q-brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s Theorem on the number of ones in all partitions of n, and provide several new combinatorial results.     

Weighted divisor sums and Bessel function series, IV

One fragment (p.?335) published with Ramanujan??s Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite series of Bessel functions. The identities are connected with the classical circle and divisor problems, respectively. This paper is devoted to the first identity. First, we obtain a generalization in the setting of Riesz sums. Second, we prove a trigonometric analogue.     

The Rogers-Ramanujan continued fraction and a new Eisenstein series identity

With two elementary trigonometric sums and the Jacobi theta function θ₁, we provide a new proof of two Ramanujan's identities for the Rogers-Ramanujan continued fraction in his lost notebook. We further derive a new Eisenstein series identity associated with the Rogers-Ramanujan continued fraction.     

On a partition identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called “Beck-type” companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.     

Duality for finite multiple harmonic q-series

We define two finite q-analogs of certain multiple harmonic series with an arbitrary number of free parameters, and prove identities for these q-analogs, expressing them in terms of multiply nested sums involving the Gaussian binomial coefficients. Special cases of these identities—for example, with all parameters equal to 1—have occurred in the literature. The special case with only one parameter reduces to an identity for the divisor generating function, which has received some attention in connection with problems in sorting theory. The general case can be viewed as a duality result, reminiscent of the duality relation for the ordinary multiple zeta function.     

Generalized Euler and Chu-Vandermonde identities

The object of this paper is to present two surprisingly general identities involving the generating functions of the number of inversions between multisets (of not necessarily uniform multiplicities). The first identity is a generalization of the Chu-Vandermonde identity, and the second a well-known identity of Euler. Our proofs are based on a one-to-one correspondence between multisubsets and certain permissible paths in a digraph with monomial weights.     

Transformation of certain bilinear generating functions

Summary Erdélyi's generalizations of the Hardy-Hille formula are extended to series involving arbitrary coefficients; see(1.6), …,(1.9) below. These identities may be thought of as identities in formal power series. The proofs are simple: each identity in shown to be equivalent to an algebraic identity that is easily verified. Supported in part by NSF grant GP-7855. Entrata in Redazione il 18 novembre 1969.     

A three-term theta function identity and its applications

In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan's lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary.     

Families of multisums as mock theta functions

In view of the Bailey lemma and the relations between Hecke-type sums and Appell–Lerch sums given by Hickerson and Mortenson, we find that many Bailey pairs given by Slater can be used to deduce mock theta functions. Therefore, by constructing generalized Bailey pairs with more parameters, we derive some new families of mock theta functions. Meanwhile, some identities between new mock theta functions and classical ones are established. Furthermore, based on the proofs of the main theorems, many q-hypergeometric transformations are obtained.     

Identities and congruences for the general partition and Ramanujan’s tau functions

We present some identities and congruences for the general partition function p _r (n). In particular, we deduce some known identities for Ramanujan’s tau function and find simple proofs of Ramanujan’s famous partition congruences for modulo 5 and 7. Our emphasis throughout this paper is to exhibit the use of Ramanujan’s theta functions to generate identities and congruences for general partition function.     

Some Fine combinatorics

In 1988, N.J. Fine published a monograph entitled Basic Hypergeometric Series and Applications in which he proved a number of results concerning the series F(a,b;t:q). In this paper, we present a new combinatorial interpretation for the series F(a,b;t:q) and use Fine’s work as a guide for proving the Rogers–Fine identity and many of its properties in this setting.     

Constant Term Methods in the Theory of Tesler Matrices and Macdonald Polynomial Operators

The Tesler matrices with hook sums (a ₁, a ₂, . . . , a _n ) are non-negative integral upper triangular matrices, whose i ^th diagonal element plus the sum of the entries in the arm of its (french) hook minus the sum of the entries in its leg is equal to a _i for all i. In a recent paper [6], the second author expressed the Hilbert series of the Diagonal Harmonic modules as a weighted sum of the family of Tesler matrices with hook weights (1, 1, . . . , 1). In this paper we use the constant term algorithm developed by the third author to obtain a Macdonald polynomial interpretation of these weighted sum of Tesler matrices for arbitrary hook weights. In particular, we also obtain new and illuminating proofs of the results in [6].     

Weighted Divisor Sums and Bessel Function Series

On page 335 in his lost notebook, Ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary Bessel functions. We provide the first published proof of this result. The identity yields as corollaries representations of weighted divisor sums, in particular, the summatory function for r₂(n), the number of representations of the positive integer n as a sum of two squares. Research partially supported by grant MDA H92830-04-1-0027 from the National Security Agency.     

Some identities on the Catalan, Motzkin and Schröder numbers

In this paper, some identities between the Catalan, Motzkin and Schröder numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schröder numbers with the Motzkin numbers, respectively.     

Combinatorial Proofs of Various q-Pell Identities via Tilings

Recently, Benjamin, Plott, and Sellers proved a variety of identities involving sums of Pell numbers combinatorially by interpreting both sides of a given identity as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes. In this article, we state and prove q-analogues of several Pell identities via weighted tilings.     

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

Let k be a positive number and t _k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t _2k (n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t ₁₂(n), t ₁₆(n), t ₂₀(n), t ₂₄(n), and t ₂₈(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t ₃₂(n). In addition, we derive some identities involving the Ramanujan function (n), the divisor function ₁₁(n), and t ₂₄(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.