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Open Access Research

A new approach to connect algebra with analysis: relationships and applications between presentations and generating functions

Ismail Naci Cangül1*, Ahmet Sinan Çevik2 and Yılmaz Şimşek3

Author affiliations

1 Department of Mathematics, Faculty of Arts and Science, Uludag University, Gorukle Campus, Bursa, 16059, Turkey

2 Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 42075, Turkey

3 Department of Mathematics, Faculty of Art and Science, Akdeniz University, Campus, Antalya, 07058, Turkey

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Citation and License

Boundary Value Problems 2013, 2013:51  doi:10.1186/1687-2770-2013-51

Published: 14 March 2013

Abstract

For a minimal group (or monoid) presentation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a>, let us suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M1">View MathML</a> satisfies the algebraic property of either being efficient or inefficient. Then one can investigate whether some generating functions can be applied to it and study what kind of new properties can be obtained by considering special generating functions. To establish that, we will use the presentations of infinite group and monoid examples, namely the split extensions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/51/mathml/M4">View MathML</a>, respectively. This study will give an opportunity to make a new classification of infinite groups and monoids by using generating functions.

MSC: 11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

Keywords:
efficiency; p-Cockcroft property; split extension; generating functions; Stirling numbers; array polynomials