On identities of the Rogers-Ramanujan type

A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization of the Rogers-Ramanujan identities (Amer. J. Math., 83, 393–399 (1961)). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater’s list (Proc. London Math. Soc. (2) 54, 147–167 (1952)), as well as the new identities presented here. A list of 26 new double sum–product Rogers-Ramanujan type identities are included as an Appendix. 2000 Mathematics Subject Classification Primary—11B65; Secondary—11P81, 05A19, 39A13     

一些Rogers-Ramanujan恒等式的广义形式

本文用简便的方法，得到了一些Ｒｏｇｅｒｓ－Ｒａｍａｎｕｊａｎ恒等式的广义形式．     

New finite Rogers-Ramanujan identities

We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson’s transformation formula by specialization or through Bailey’s method, the second similar formula can be proved either by using the first formula and the q-Gosper algorithm, or through the so-called Bailey lattice.      

Some more identities of the Rogers-Ramanujan type

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. The research of the first author was partially supported by National Science Foundation grant DMS-0300126.     

A partition bijection related to the Rogers-Selberg identities and Gordon's theorem

We provide a bijective map from the partitions enumerated by the series side of the Rogers-Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers-Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.     

Multiple partitions, lattice paths and a Burge-Bressoud-type correspondence

A bijection is presented between (1): partitions with conditions f_j+f_j+1≤k−1 and f₁≤i−1, where f_j is the frequency of the part j in the partition, and (2): sets of k−1 ordered partitions (n⁽¹⁾,n⁽²⁾,…,n^(k−1)) such that and , where m_j is the number of parts in n^(j). This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the k−1 ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud’s version of the Burge correspondence.     

Rogers-Ramanujan Type Identities for Partitions with Attached Odd Parts

In this paper we present a new infinite family of partition identities. The genesis of our work lies in two formulas of Lucy Slater related to the modulus 8. Hirschhorn, Agarwal and Subbarao have previously found intriguing interpretations for Slater's formula, but none has led to an infinite family of results.     

The multisection method for triple products and identities of Rogers-Ramanujan type

By applying the bisection and trisection method to Jacobi's triple product identity, we establish several identities factorizing sum and difference of infinite products, which lead, in turn, to new and elementary proofs for twenty identities of Rogers-Ramanujan type.     

Cumulants, lattice paths, and orthogonal polynomials

A formula expressing free cumulants in terms of Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and the Lagrange inversion formula. For the converse we discuss Gessel–Viennot theory to express Hankel determinants in terms of various cumulants.     

A combinatorial proof of the Rogers-Ramanujan and Schur identities

We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.     

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.     

The number of lattice paths below a cyclically shifting boundary

We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical “reflection” argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special “staircases.”     

New proofs of identities of Lebesgue and Göllnitz via tilings

In 1840, V.A. Lebesgue proved the following two series-product identities:

     

The genesis of involutions (Polarizations and lattice paths)

The number of Borel orbits in the symmetric space $S{L}_n∕S(G{L}_p\times G{L}_q)$ is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored.     

A class of combinatorial identities

A general theorem for providing a class of combinatorial identities where the sum is over all the partitions of a positive integer is proven. Five examples as the applications of the theorem are given.     

On overpartition pairs into odd parts modulo powers of 2

In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for ${\overline{pp}}_o\left(n\right)$, the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer $k$, ${\overline{pp}}_o\left(n\right)$ is almost always divisible by $2^k$. In this paper, we prove several congruences for ${\overline{pp}}_o\left(n\right)$ modulo higher powers of 2 in an elementary way.     

Supernomial Coefficients,Polynomial Identities and q-Series

q-Analogues of the coefficients of xa in the expansion of _j=1 ^N (1 + x + + x^j)^Lj are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the q-supernomial coefficients are derived, and a combinatorial interpretation using generalized Durfee dissection partitions is given. Polynomial identities of boson-fermion-type, based on the continued fraction expansion of p/k and involving the q-supernomial coefficients, are proven. These include polynomial analogues of the Andrews-Gordon identities. Our identities unify and extend many of the known boson-fermion identities for one-dimensional configuration sums of solvable lattice models, by introducing multiple finitization parameters.     

Lattice paths and generalized cluster complexes

In this paper we propose a variant of the generalized Schröder paths and generalized Delannoy paths by giving a restriction on the positions of certain steps. This generalization turns out to be reasonable, as attested by the connection with the faces of generalized cluster complexes of types A and B. As a result, we derive Krattenthaler's F-triangles for these two types by a combinatorial approach in terms of lattice paths.     

Limits of areas under lattice paths

We consider sequences of polynomials which count lattice paths by area. In some cases the reversed polynomials approach a formal power series as the length of the paths tend to infinity. We find the limiting series for generalized Schröder, Motzkin, and Catalan paths. The limiting series for Schröder paths and their generalizations are shown to count partitions with restrictions on the multiplicities of odd parts and no restrictions on even parts. The limiting series for generalized Motzkin and Catalan paths are shown to count generalized Frobenius partitions and some related arrays.     

Partially directed paths in a wedge

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional equation for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by Y=±pX, and an asymmetric wedge defined by the lines Y=pX and Y=0, where p>0 is an integer. We prove that the growth constant for all these models is equal to , independent of the angle of the wedge. We derive function equations for both models, and obtain explicit expressions for the generating functions when p=1. From these we find asymptotic formulas for the number of partially directed paths of length n in a wedge when p=1.The functional equations are solved by a variation of the kernel method, which we call the “iterated kernel method.” This method appears to be similar to the obstinate kernel method used by Bousquet-Mélou (see, for example, references [M. Bousquet-Mélou, Counting walks in the quarter plane, in: Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, Trends Math., Birkhäuser, 2002, pp. 49-67; M. Bousquet-Mélou, Four classes of pattern-avoiding permutations under one roof: Generating trees with two labels, Electron. J. Combin. 9 (2) (2003) R19; M. Bousquet-Mélou, M. Petkovšek, Walks confined in a quadrant are not always D-finite, Theoret. Comput. Sci. 307 (2) (2003) 257-276]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi θ-functions, and have natural boundaries on the unit circle.