Parity in partition identities

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f ₀(q) along with a new proof of the Hecke-type series representation.     

New series representations for Jacobi?s triple product identity and more via the q-Markov method

We derive new series representations for Jacobi?s triple product identity, the q-binomial theorem, q-analogs of the exponential function, and more with several special cases using the q-Markov-WZ method.     

Bender's generalizedq-binomial Vandermonde convolution

A simple arithmetical proof and a generalization of Bender's generalizedq-binomial Vandermonde convolution are given.     

Generalizing inplace multiplicity identities for integer compositions

In a recent paper, the authors gave two new identities for compositions, or ordered partitions, of integers. These identities were based on closely-related integer partition functions which have recently been studied. In the process, we also extensively generalized both of these identities. Since then, we asked whether one could generalize one of these results even further by considering compositions in which certain parts could come from t kinds (rather than just two kinds, which was the crux of the original result). In this paper, we provide such a generalization. A straightforward bijective proof is given and generating functions are provided for each of the types of compositions which arise. We close by briefly mentioning some arithmetic properties satisfied by the functions which count such compositions.     

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite difference equations with random, possibly non symmetric coe?cients. Under the assumption that the coe?cients are stationary and ergodic in the quantitative form of a logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d≥2. Similar estimates have recently been obtained in the special case of diagonal coe?cients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green’s function in weighted spaces. In the critical case d = 2, our argument for the estimate on the gradient of the elliptic Green’s function uses a Calderón–Zygmund estimate in discrete weighted spaces, which we state and prove. As applications, we provide a quantitative two-scale expansion and a quantitative approximation of the homogenized coe?cients.     

Undetected error probability of q-ary constant weight codes

In this paper, we introduce a new combinatorial invariant called q-binomial moment for q-ary constant weight codes. We derive a lower bound on the q-binomial moments and introduce a new combinatorial structure called generalized (s, t)-designs which could achieve the lower bounds. Moreover, we employ the q-binomial moments to study the undetected error probability of q-ary constant weight codes. A lower bound on the undetected error probability for q-ary constant weight codes is obtained. This lower bound extends and unifies the related results of Abdel-Ghaffar for q-ary codes and Xia-Fu-Ling for binary constant weight codes. Finally, some q-ary constant weight codes which achieve the lower bounds are found.      

The q-Markov-WZ Method

Andrei Markov’s 1890 beautiful ad-hoc method of transforming a series of hypergeometric type into a rapidly-converging series was upgraded recently to a full-fledged method by Mohammed and Zeilberger, but only for the ordinary case. In this article, the q-case is developed and it is shown how Markov’s ad-hoc method, when coupled with q-WZ theory and q-Gosper’s algorithm, leads to a new class of identities and very fast convergence-acceleration series that can be applied to any infinite series of q-hypergeometric type.Received August 27, 2004     

q-Analogs of some congruences involving Catalan numbers

We present some variations on the Greene–Krammer?s identity which involve q-Catalan numbers. Our method reveals an intriguing analogy between these new identities and some congruences modulo a prime.     

Pseudo q-Engel expansions and Rogers-Ramanujan type identities

Abstract

Andrews, Knopfmacher and Knopfmacher have used the Schur polynomials to consider the celebrated Rogers-Ramanujan identities in the context of q-Engel expansions. We extend this view using similar polynomials, provided by Sills, in the context of Slater's list of 130 Rogers-Ramanujan type identities.     

Généralisation d'identités de Carlitz,Howard et Lehmer

Divided differences provide an efficient method for computing with functions of several variables.In this note, we use them to generalize the Newton interpolation formula, and obtain an orthogonality relation (3.3). From this, we deduce two inversion formulas (3.4) and (3.8) involving two infinite sets of variables.The generating functions (1.1) to (1.4) of Carlitz and Howard are obtained by a mere specialization of variables in the preceding inversion formulas.As an illustration, we show how to recover several identities due to Carlitz and Lehmer, and we give a newq-analog of the generating series of the Howard numbers (formulas 4.22 and 4.23).

     

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.     

Rank and Conjugation for the Frobenius Representation of an Overpartition

We discuss conjugation and Dyson’s rank for overpartitions from the perspective of the Frobenius representation. More specifically, we translate the classical definition of Dyson’s rank to the Frobenius representation of an overpartition and define a new kind of conjugation in terms of this representation. We then use q-series identities to study overpartitions that are self-conjugate with respect to this conjugation. Received June 28, 2004     

On identities of the Rogers-Ramanujan type

An elementary approach to a number of identities of the Rogers-Ramanujan type is given. It is shown that analytic formulas like, e.g., the Rogers-Ramanujan, the Rogers-Selberg and the Göllnitz-Gordon identities can be obtained essentially as consequences of the q-binomial theorem and the q-Vandermonde formula.     

An overpartition analogue of partitions with bounded differences between largest and smallest parts

We study the generating function for overpartitions with bounded differences between largest and smallest parts, which is analogous to a result of Breuer and Kronholm on integer partitions. We also connect this problem with over $q$-binomial coefficients introduced by Dousse and Kim.     

Bit strings withoutq-separation

LetB(n, q) denote the number of bit strings of lengthn withoutq-separation. In a bit string withoutq-separation no two 1's are separated by exactlyq – 1 bits.B(n, q) is known to be expressible in terms of a product of powers of Fibonacci numbers. Two new and independent proofs are given. The first proof is by combinatorial enumeration, while the second proof is inductive and expressesB(n, q) in terms of a recurrence relation.     

q-Pell sequences and two identities of V.A. Lebesgue

We examine a pair of Rogers-Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities.     

On factorization of polynomials in henselian valued fields

Guàrdia, Montes and Nart generalized the well-known method of Ore to find complete factorization of polynomials with coe?cients in finite extensions of p-adic numbers using Newton polygons of higher order (cf. [Trans. Amer. Math. Soc. 364 (2012), 361–416]). In this paper, we develop the theory of higher order Newton polygons for polynomials with coe?cients in henselian valued fields of arbitrary rank and use it to obtain factorization of such polynomials. Our approach is different from the one followed by Guàrdia et al. Some preliminary results needed for proving the main results are also obtained which are of independent interest.     

A Double Bounded Version of Schur's Partition Theorem

Dedicated to the memory of Paul Erdős Schur's partition theorem states that the number of partitions of n into distinct parts (mod 3) equals the number of partitions of n into parts which differ by 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts (mod 3) and another on the parts (mod 3), and by keeping track of the number of parts in each of the residue classes (mod 3). Despite the long history of Schur's theorem, our result is new, and extends earlier work of Andrews, Alladi-Gordon and Bressoud. We give combinatorial and q-theoretic proofs of our result. The special case L=M leads to a representation of the generating function of the underlying partitions in terms of the q-trinomial coefficients extending a similar previous representation of Andrews. Received November 18, 1999 Research of the first author supported in part by NSF Grant DMS-0088975.     

The homogeneous q-difference operator

We introduce a q-differential operator D_xy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.     

Another homogeneous q-difference operator

In this paper we show the equivalence between Goldman-Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers-Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers-Szegö polynomials. We give a new formula for the homogeneous Rogers-Szegö polynomials h_n(x,y|q). We introduce a q-difference operator θ_xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659-668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s_n(x,y;b|q) and derive their generating function by using the new homogeneous q-shift operator L(bθ_xy).