# American Institute of Mathematical Sciences

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

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

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

Received  August 2010 Revised  March 2011 Published  October 2011

We present a new approach to exponential functions on time scales and to timescale analogues of ordinary di erential 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 Padé approximants, we obtain Pade-exponential functions. Moreover, the exact exponential function on time scales is de fined. Finally, we present applications of the Cayley-exponential function in the $q$-calculus and suggest a general approach to dynamic systems on Lie groups.
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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