On the Exponential and Trigonometric \(q,\omega \)-Special Functions
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The purpose of this article is to continue the study of \(q,\omega \)-special functions in the spirit of Wolfgang Hahn from the previous papers by Annaby et al. and Varma et al. By introducing the new variable \(\omega \), we develop a quite similar calculus consisting of two dual exponential, hyperbolic and trigonometric functions. The concept even and odd functions is replaced by \(x,\omega \)-even and odd, since a change of sign in x is always accompanied by a change of sign in \(\omega \). In the same way, formulas for chain rule, Leibniz theorem, \(q,\omega \)-additions for the three above functions are introduced. Graphs for these functions are shown, which closely resemble the original ones. To enable trigonometric formulas with half argument and de Moivre theorem, Ward numbers and \(q,\omega \)-rational numbers are introduced.
Keywordsq, \(\omega \)-special functions q, \(\omega \)-difference operator q, \(\omega \)-rational number Similar graphs Rules for zeros
Mathematics Subject Classification (2010)05A30
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