Abstract: Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas’ theorem on binomial coefficients modulo p not only extends naturally to the case of negative entries, but even to the Gaussian case. PubDate: 2019-11-16

Abstract: We introduce a polynomial E(d, t, x) in three variables that comes from the intersections of a family of ellipses described by Euler. For fixed odd integers \(t\ge 3\), the sequence of E(d, t, x) with d running through the integers produces, conjecturally, sequences of “twin composites” analogous to the twin primes of the integers. This polynomial and its lower degree relative R(d, t, x) have strikingly simple discriminants and resolvents. Moreover, the roots of R for certain values of d have continued fractions with at least two large partial quotients, the second of which mysteriously involves the 12th cyclotomic polynomial. Various related polynomials whose roots also have conjecturally strange continued fractions are also examined. PubDate: 2019-11-16

Abstract: We establish an algorithm for producing formulas for p(n, m, N), the function enumerating partitions of n into at most m parts with no part larger than N. Recent combinatorial results of H. Hahn et al. on a collection of partition identities for p(n, 3, N) are considered. We offer direct proofs of these identities and then place them in a larger context of the unimodality of Gaussian polynomials \(N+m\brack m\) whose coefficients are precisely p(n, m, N). We give complete characterizations of the maximal coefficients of \({M\brack 3}\) and \({M\brack 4}\). Furthermore, we prove a general theorem on the period of quasipolynomials for central/maximal coefficients of Gaussian polynomials. We place some of Hahn’s identities into the context of some known results on differences of partitions into at most m parts, p(n, m), which we then extend to p(n, m, N). PubDate: 2019-11-15

Abstract: Given an integer base \(b\ge 2\), we investigate a multivariate b-ary polynomial analogue of Stern’s diatomic sequence which arose in the study of hyper b-ary representations of integers. We derive various properties of these polynomials, including a generating function and identities that lead to factorizations of the polynomials. We use some of these results to extend an identity of Courtright and Sellers on the b-ary Stern numbers \(s_b(n)\). We also extend a result of Defant and a result of Coons and Spiegelhofer on the maximal values of \(s_b(n)\) within certain intervals. PubDate: 2019-11-15

Abstract: We consider certain classes of compositions of numbers based on the recently introduced extension of conjugation to higher orders. We use generating functions and combinatorial identities to provide enumeration results for compositions possessing conjugates of a given order. Working under some popular themes in the theory, we show that results for these compositions specialize to standard results in a natural way. We also give a generalization of MacMahon’s identities for inverse-conjugate compositions and discuss inverse-reciprocal compositions. PubDate: 2019-11-15

Abstract: We present an infinite family of Borwein type \(+ - - \) conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument. PubDate: 2019-11-13

Abstract: Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely, (K, i)-singular overpartitions for integers K, i with \( 1\le i<K/2\). The definition of such singular overpartitions requires successive ranks, parity blocks and anchors. The concept of successive ranks was extensively generalized to hook differences by Andrews, Baxter, Bressoud, Burge, Forrester and Viennot in 1987. In this paper, employing hook differences, we generalize parity blocks. Using this combinatorial concept, we define \((K,i,\alpha , \beta )\)-singular overpartitions for positive integers \(\alpha , \beta \) with \(\alpha +\beta <K\), and then we show some connections between such singular overpartitions and ordinary partitions. PubDate: 2019-11-13

Abstract: We give a simple and a more explicit proof of a mod 4 congruence for a series involving the little q-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function. PubDate: 2019-11-11

Abstract: We discuss q-analogues of the classical congruence \(\left( {\begin{array}{c}ap\\ bp\end{array}}\right) \equiv \left( {\begin{array}{c}a\\ b\end{array}}\right) \pmod {p^3}\), for primes \(p>3\), as well as its generalisations. In particular, we prove related congruences for (q-analogues of) integral factorial ratios. PubDate: 2019-11-11

Abstract: In this paper, we generalize the Andrews–Yee identities associated with the third-order mock theta functions \(\omega (q)\) and \(\nu (q)\). We obtain some q-series transformation formulas, one of which gives a new Bailey pair. Using the classical Bailey lemma, we derive a product formula for two \({}_2\phi _1\) series. We also establish recurrence relations and transformation formulas for two finite sums arising from the Andrews–Yee identities. PubDate: 2019-11-09

Abstract: Usually, the Weierstraß gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstraß gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann–Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case when the given compact Riemann surface is associated with the modular curve \(X_0(N)\). PubDate: 2019-11-07

Abstract: MacMahon showed that the generating function for partitions into at most k parts can be decomposed into a partial fraction-type sum indexed by the partitions of k. In the present work, a generalization of MacMahon’s result is given, which in turn provides a full combinatorial explanation. PubDate: 2019-11-05

Abstract: In his lost notebook, Ramanujan listed five identities related to the false theta function: $$\begin{aligned} f(q)=\sum _{n=0}^\infty (-1)^nq^{n(n+1)/2}. \end{aligned}$$ A new combinatorial interpretation and a proof of one of these identities are given. The methods of the proof allow for new multivariate generalizations of this identity. Additionally, the same technique can be used to obtain a combinatorial interpretation of another one of the identities. PubDate: 2019-11-02

Abstract: We give simple proofs of Hecke–Rogers indefinite binary theta series identities for the two Ramanujan’s fifth order mock theta functions \(\chi _0(q)\) and \(\chi _1(q)\) and all three of Ramanujan’s seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related. PubDate: 2019-11-01

Abstract: In this note, we show how to use cylindric partitions to rederive the four \(A_2\) Rogers–Ramanujan identities originally proven by Andrews, Schilling and Warnaar, and provide a proof of a similar fifth identity. PubDate: 2019-10-31

Abstract: Recently, Sun (Two q-analogues of Euler’s formula \(\zeta (2)=\pi ^2/6\) . arXiv:1802.01473, 2018) obtained q-analogues of Euler’s formula for \(\zeta (2)\) and \(\zeta (4)\) . Sun’s formulas were based on identities satisfied by triangular numbers and properties of Euler’s q-Gamma function. In this paper, we obtain a q-analogue of \(\zeta (6)=\pi ^6/945\) . Our main results are stated in Theorems 2.1 and 2.2 below. PubDate: 2019-10-30

Abstract: 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 identity. Various other general identities, for example certain expansions of \((q;q)_{\infty }\) and \((q;q)_{\infty }^k\) , \(k\ge 3\) , as combinations of Jacobi triple products, are also proved. PubDate: 2019-10-30

Abstract: We use the q-binomial theorem to prove three new polynomial identities involving q-trinomial coefficients. We then use summation formulas for the q-trinomial coefficients to convert our identities into another set of three polynomial identities, which imply Capparelli’s partition theorems when the degree of the polynomial tends to infinity. This way we also obtain an interesting new result for the sum of the Capparelli’s products. We finish this paper by proposing an infinite hierarchy of polynomial identities. PubDate: 2019-10-29

Abstract: For an integer \(m\ge 2\) , a partition \(\lambda =(\lambda _1,\lambda _2,\ldots )\) is called m-falling, a notion introduced by Keith, if the least non-negative residues mod m of \(\lambda _i\) ’s form a nonincreasing sequence. We extend a bijection originally due to the third author to deduce a lecture hall theorem for such m-falling partitions. A special case of this result gives rise to a finite version of Pak–Postnikov’s (m, c)-generalization of Euler’s theorem. Our work is partially motivated by a recent extension of Euler’s theorem for all moduli, due to Xiong and Keith. We note that their result actually can be refined with one more parameter. PubDate: 2019-10-29

Abstract: We show that there is a concept of q-translation behind the approach used by Liu to prove summation and transformation identities for q-series. We revisit the q-translation associated with the Askey–Wilson operator introduced in Ismail (Ann Comb 5(3–4):347–362, 2001), simplify its formalism and point out new properties of this translation operator. PubDate: 2019-10-22