On a certain weighted partition function

compactifications and function spaces on weighted semigruops

chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...

A Crank Analog on a Certain Kind of Partition Function Arising from the Cubic Continued Fraction

In a series of papers, H.-C. Chan has studied congruence properties of a certain kind of partition function that arises from Ramanujan’s cubic continued fraction. This partition function a(n), is defined by ∑∞ n=0 a(n)q n = 1 (q;q)∞(q2;q2)∞ . In particular, he proved that a(3n + 2) ≡ 0 (mod 3). As Chan mentioned in his paper, it is natural to ask if there exists an analog of the rank or the cra...

On a Multiplicative Partition Function

Let D(s) = ∑∞ m=1 amm −s be the Dirichlet series generated by the infinite product ∏∞ k=2(1− k−s). The value of am for squarefree integers m with n prime factors depends only on the number n, and we let f(n) denote this value. We prove an asymptotic estimate for f(n) which allows us to solve several problems raised in a recent paper by M. V. Subbarao and A. Verma.

On Stanley's Partition Function

Stanley defined a partition function t(n) as the number of partitions λ of n such that the number of odd parts of λ is congruent to the number of odd parts of the conjugate partition λ modulo 4. We show that t(n) equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p(n)− t(n). As a consequen...

Sums of an Entire Function in Certain Weighted L-spaces