Abstract
In a recent joint paper (Cevik et al. in Hacet. J. Math. Stat., acceptted), the authors have investigated the p-Cockcroft property (or, equivalently, efficiency) for a presentation, say , of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Moreover, they have presented sufficient conditions on a special case for to be minimal whilst it is inefficient. In this paper, by considering these results, we first show that the presentations of the form can actually be represented by characteristic polynomials. After that, some connections between representative characteristic polynomials and generating functions in terms of array polynomials over the presentation will be pointed out. Through indicated connections, the existence of an equivalence among each generating function in itself is claimed studied in this paper. MSC:11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.
Highlights
Introduction and preliminariesIn a recently published joint paper [ ], the authors computed algebraic relations in terms of Ramanujan-Eisenstein series
There are so many similar studies about figuring out the relationship between algebraic relations and special generating functions in the literature, we have not seen any such studies of relationship between presentations and generating functions
In terms of efficiency and inefficiency over group and monoid presentations, very important characterizations are given for related algebraic structures
Summary
It is strictly referred to [ – ] for fundamentals and properties of the algebraic subject used in this subsection. We will mainly present the efficiency (equivalently, p-Cockcroft property for a prime p) for the semi-direct product of free abelian monoid K having rank two by a finite cyclic monoid A of order k. Let us suppose that the kth (k ∈ Z+) power of M is defined as. The subsets C and C contain the generating pictures PS,x (which contains a non-spherical subpicture BS,x as depicted in [ ]), PR,y and PR,y of the trivializer set XE These pictures can be presented as in Figure (a) and (b). Suppose that the positive integer d, defined in ( ), is equal to a prime p such that p | (k –l). Proposition ([ ]) The presentation PE in ( ) is minimal but inefficient if p is an odd prime and either
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