Restricted partition functions as Bernoulli and Eulerian polynomials of higher order

Explicit expressions for restricted partition function W(s,d^m) and its quasiperiodic components W_j(s,d^m) (called Sylvester waves) for a set of positive integers d^m = {d₁, d₂, ..., d_m} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Eulerian polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of finite groups is discussed. 2000 Mathematics Subject Classification Primary—11P81; Secondary—11B68, 11B37 The research was supported in part (LGF) by the Gileadi Fellowship program of the Ministry of Absorption of the State of Israel.     

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.     

Some new identities for colored partition

In this paper, we discover several new identities on colored partitions and provide proofs for them. Many colored partition identities presented in the paper do not belong to the general and unified combinatorial framework provided by Sandon and Zanello. Most of our proofs depend upon new modular equations given by employing the method of reciprocation.     

Nontrivial identities in finitely generated semigroups

Ivanovo. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 3, pp. 191–197, May–June, 1989.     

Ramanujan type identities and congruences for partition pairs

Using elementary methods, we establish several new Ramanujan type identities and congruences for certain pairs of partition functions.     

On identities of finite involution semigroups

We investigate the question of whether a finite involution semigroup is inherently nonfinitely based (INFB), which means that it is not contained in any finitely based locally finite variety. Although we fall short of a full characterization, we nevertheless clarify a number of interesting subcases.     

Decompositions of semigroups induced by identities

In this paper we consider decompositions of semigroups induced by identities. Here we give some new characterizations of a semilattice of Archimedean semigroups and, using this, we describe all identities which induce decompositions into a semilattice of Archimedean semigroups. Also, we give a solution for one problem ofШеврин andСуханов [27]. Supported by Grant 0401A of RFNS through Math. Inst. SANU     

New analogues of Ramanujan's partition identities

We establish several new analogues of Ramanujan's exact partition identities using the theory of modular functions.     

Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities

The Ramanujan Journal - Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five...     

Differentiation identities for hypergeometric functions

It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions have been used widely in various fields of applied mathematics and natural sciences. In this expository note, we provide a simple proof of the differentiation identities, which is based only on the definition of the coefficients for the power series expansion of the hypergeometric functions.     

Determinant identities for theta functions

Two proofs of a theta function identity of R.W. Gosper and R. Schroeppel are given. A cubic analogue is presented, and several interesting special cases are noted.     

Generalizations of the Bellavitis partition identities

Let p = p(a, b, c) be the number of partitions of a into b parts, no part exceeding c. Bellavitis and perhaps some earlier writers noted that p satisfies three very simple identities. Here p is generalized to a function of k + 1 variables in a natural way. One of the identities then generalizes; the proof of this (which depends on the P. Hall commutator collecting process) is given only for k = 3.     

A variety of semigroups without an irreducible basis for identities

An example of a variety of semigroups not having an irreducible basis for identities is given. Also a variety of semigroups, the lattice of subvarieties of which contains a nonidentity element without any coverings, is constructed.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 865–872, June, 1977.     

On identities in groups of fractions of cancellative semigroups

To solve two problems of Bergman stated in 1981, we construct a group such that contains a free noncyclic subgroup (hence, satisfies no group identity) and , as a group, is generated by its subsemigroup that satisfies a nontrivial semigroup identity.

     

The Frobenius problem for numerical semigroups

In this paper, we characterize those numerical semigroups containing 〈n₁,n₂〉. From this characterization, we give formulas for the genus and the Frobenius number of a numerical semigroup. These results can be used to give a method for computing the genus and the Frobenius number of a numerical semigroup with embedding dimension three in terms of its minimal system of generators.