A four-parameter partition identity

We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrews in which he considers the generating function for partitions with respect to size, number of odd parts, and number of odd parts of the conjugate. 2000 Mathematics Subject Classification Primary—05A17; Secondary—11P81     

Singular values of the Rogers-Ramanujan continued fraction

Let z∊ C be imaginary quadratic in the upper half plane. Then the Rogers-Ramanujan continued fraction evaluated at q = e ^{2π iz} is contained in a class field of Q(z). Ramanujan showed that for certain values of z, one can write these continued fractions as nested radicals. We use the Shimura reciprocity law to obtain such nested radicals whenever z is imaginary quadratic. 2000 Mathematics Subject Classification Primary—11Y65; Secondary—11Y40     

Partition bijections, a survey

We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises. 2000 Mathematics Subject Classification Primary—05A17; Secondary—05A30; 11P83 The author was partially supported by NSA and NSF grants.     

Mixed moments of L-functions

In this paper we will study the main term of mixed moments of L-functions associated to holomorphic cusp forms of integral weight on congruence subgroups of odd and squarefree level. 2000 Mathematics Subject Classification Primary—11F67, 11M41, 11F30     

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     

Finite polynomial orbits in quadratic rings

All finite orbits of polynomial mappings in one variable in rings of integers of quadratic number fields are determined. The largest such orbit has seven elements and lies in the third cyclotomic field. 2000 Mathematics Subject Classification Primary—11R09, 11R11, 37F10     

On formulas of Ramanujan and Evans

We give short elementary proofs of a formula of Ramanujan as interpreted by Bradley, and a companion formula originally proved by W. Chu. We also give an elementary proof of a generalization of an identity originally proved using modular functions and used to study a generating function for the number of partitions with specified crank. Research partially supported by NSF grant DMS 99-70865. 2000 Mathematics Subject Classification Primary—39A70, 11P83; Secondary—33C20     

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series. 2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15 The work of the second author is supported by an Alexander von Humboldt research fellowship Dedication: To Leonhard Euler on his 300th birthday.     

On the number of primitive representations of integers as sums of squares

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15, 33E05.     

Mixed Volume Computation for Semi-Mixed Systems

Abstract. The Li—Li algorithm produced in [11] for the mixed volume computation of fully mixed polynomial systems is reconstructed in this article for general semi-mixed polynomial systems. Taking the special structure of the semi-mixed supports of the systems into account, the resulting algorithm, illustrated by numerical results, can dramatically speed up the mixed volume computation, especially when the systems are unmixed. Even when applied to fully mixed systems, the new algorithm improves the speed of the Li—Li algorithm by a considerable amount.     

Spaces of modular forms generated by eta-quotients

In this paper, we study the spaces of modular forms on Γ₀(N) generated by eta-quotients, where the genus of Γ₀(N) is zero or N is a prime. Our results give an answer to a question of Ono (2004, Problem 1.68). 2000 Mathematics Subject Classification Primary—11F20, 11F11     

Mixed Volume Computation for Semi-Mixed Systems

   Abstract. The Li—Li algorithm produced in [11] for the mixed volume computation of fully mixed polynomial systems is reconstructed in this article for general semi-mixed polynomial systems. Taking the special structure of the semi-mixed supports of the systems into account, the resulting algorithm, illustrated by numerical results, can dramatically speed up the mixed volume computation, especially when the systems are unmixed. Even when applied to fully mixed systems, the new algorithm improves the speed of the Li—Li algorithm by a considerable amount.     

Hypergeometric functions and elliptic curves

We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively. 2000 Mathematics Subject Classification Primary—11G05, 33C05 This research was supported by K. Ono’s NSF grant     

Primes in progressions to moduli with a large power factor

On average, primes are uniformly distributed in short arithmetic progressions whose moduli may be divisible by high-powers of a given integer. In celebration of the seventieth birthday of Richard Askey. 2000 Mathematics Subject Classification Primary—11N13     

An identity of Andrews, multiple integrals, and very-well-poised hypergeometric series

We give a new proof of a theorem of Zudilin that equates a very-well-poised hypergeometric series and a particular multiple integral. This integral generalizes integrals of Vasilenko and Vasilyev which were proposed as tools in the study of the arithmetic behaviour of values of the Riemann zeta function at integers. Our proof is based on limiting cases of a basic hypergeometric identity of Andrews. Dedicated to Richard Askey on the occasion of his 70th birthday. Research partially supported by the programme “Improving the Human Research Potential” of the European Commission, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”. 2000 Mathematics Subject Classification Primary—33C20; Secondary—11J72     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Multiplicative character sums with the sum of g-ary Digits Function

We establish upper bounds for multiplicative character sums with the function σ_g (n) which computes the sum of the digits of n in a fixed base g ≥ 2. Our results may be viewed as analogues of some previously known results for exponential sums with sum of g-ary digits function. 2000 Mathematics Subject Classification Primary—11A63, 11B50, 11L40     

An Euler product transform applied to q-series

This paper introduces the concept of a D-analogue. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series, often called the basic hypergeometric series. The main result in this paper is a transform, based on an Euler product over the primes. Examples given are D-analogues of the q-binomial theorem and the q-Gauss summation. 2000 Mathematics Subject Classification Primary—11M41; Secondary—33D15, 30B50     

q-Series and L-functions related to half-derivatives of the Andrews-Gordon identity

Studied is a generalization of Zagier’s q-series identity. We introduce a generating function of L-functions at non-positive integers, which is regarded as a half-differential of the Andrews-Gordon q-series. When q is a root of unity, the generating function coincides with the quantum invariant for the torus knot. 2000 Mathematics Subject Classification Primary—11F67, 57M27, 05A30, 11F23     

A note on slant submanifolds of nearly trans-Sasakian manifolds

The geometry of slant submanifolds of a nearly trans-Sasakian manifold is studied when the tensor field Q is parallel. It is proved that Q is not parallel on the submanifold unless it is anti-invariant and thus the result of [CABRERIZO, J. L.—CARRIAZO, A.—FERNANDEZ, L. M.—FERNANDEZ, M.: Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (2000), 125–138] and [GUPTA, R. S.—KHURSHEED HAIDER, S. M.—SHARFUDIN, A.: Slant submanifolds of a trans-Sasakian manifold, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 47 (2004), 45–57] are generalized.