Two Meromorphic Functions Share Some Pairs of Small Functions

In this article, we prove that two meromorphic functions $f$ and $g$ must be linked by a quasi-Möbius transformation if they share two pairs of small functions ignoring multiplicities and share other four small functions with multiplicities truncated by 2. We also show that two meromorphic functions which share seven pairs of small functions ignoring multiplicities are linked by a quasi-Möbius transformation.     

On the Stirling numbers associated with the meromorphic Weyl algebra

The generalized Stirling numbers introduced recently (Mansour and Schork, 2011 [5], Mansour et al., 2011 [6]) are considered in detail for the particular case $s=2$ corresponding to the meromorphic Weyl algebra. A combinatorial interpretation in terms of perfect matchings is given for these meromorphic Stirling numbers and the connection to Bessel functions is discussed. Furthermore, two related $q$-polynomial identities are derived.     

Quadratic forms and congruences for ℓ-regular partitions modulo 3, 5 and 7

Let $b_{\ell }(n)$ be the number of ℓ-regular partitions of n. We show that the generating functions of $b_{\ell }(n)$ with $\ell =3,5,6,7$ and 10 are congruent to the products of two items of Ramanujan's theta functions $\psi (q)$, $f(-q)$ and ${(q;q)}_{\infty }^3$ modulo 3, 5 and 7. So we can express these generating functions as double summations in q. Based on the properties of binary quadratic forms, we obtain vanishing properties of the coefficients of these series. This leads to several infinite families of congruences for $b_{\ell }(n)$ modulo 3, 5 and 7.     

Lehmer's totient problem over Fq[x]

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x]$, where ${\mathbb{F}}_q$ is a finite field with q elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}?1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3$, $p(x)$ is the product of any 2 non-associate irreducibles of degree 1; or (iii) $q=2$, $p(x)$ is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.     

On properties and applications of (p,q)-extended τ-hypergeometric functions

We introduce the $(p,q)$-extended τ-hypergeometric and confluent hypergeometric functions along with their integral representations. We also present closed integral expressions for the Mathieu-type a-series and for the associated alternating versions whose terms contain the $(p,q)$-extended τ-hypergeometric functions with related contiguous functional relations.     

Two-sided ideals in q-deformed Heisenberg algebras

In this article, the structure of two-sided ideals in the $q$-deformed Heisenberg algebras defined by the $q$-deformed Heisenberg canonical commutation relation$\mathit{AB}-\mathit{qBA}=I$is investigated. We show that these algebras are simple if and only if $q=1$. For $q\ne 1,0$ we present an infinite descending chain of non-trivial two-sided ideals, thus deducing by explicit construction that the $q$-deformed Heisenberg algebras are not just non-simple but also non-artinian for $q\ne 1,0$. We establish a connection between the quotients of the $q$-deformed Heisenberg algebras by these ideals and the quotients of the quantum plane. We also present a number of reordering formulae in $q$-deformed Heisenberg algebras, investigate properties of deformed commutator mappings, show their fundamental importance for investigation of ideals in $q$-deformed Heisenberg algebras, and demonstrate how to apply these results to the investigation of faithfulness of representations of $q$-deformed Heisenberg algebras.