Some implications of a new approach to exponential functions on time scales

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2011, 2011(Special): 302-311. doi: 10.3934/proc.2011.2011.302

Some implications of a new approach to exponential functions on time scales

Jan L. Cieśliński ^1,

1.	Uniwersytet w Białymstoku, Wydział Fizyki, ul. Lipowa 41, 15-424 Białystok, Poland

Received August 2010 Revised March 2011 Published October 2011

Abstract
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Under a Creative Commons license

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary dierential equations. We describe in detail the Cayley-exponential function and associated trigonometric and hyperbolic functions. We show that the Cayley-exponential is related to implicit midpoint and trapezoidal rules, similarly as delta and nabla exponential functions are related to Euler numerical schemes. Extending these results on any Padé approximants, we obtain Pade-exponential functions. Moreover, the exact exponential function on time scales is defined. Finally, we present applications of the Cayley-exponential function in the $q$-calculus and suggest a general approach to dynamic systems on Lie groups.

Keywords: Cayley transformation, hyperbolic functions, Time scales, $q$-exponential function, exact discretization, first and second order dynamic equations, trigonometric functions, exponential function, Padé approximation, $q$-trigonometric functions.

Mathematics Subject Classification: Primary: 33B10, 26E70; Secondary: 34N05, 65L12.

Citation: Jan L. Cieśliński. Some implications of a new approach to exponential functions on time scales. Conference Publications, 2011, 2011 (Special) : 302-311. doi: 10.3934/proc.2011.2011.302

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