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Some implications of a new approach to exponential functions on time scales
Jan L. Cieśliński
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 Padé approximants, we obtain Pade-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.
keywords: Cayley transformation hyperbolic functions Time scales $q$-exponential function exact discretization fi rst and second order dynamic equations trigonometric functions exponential function Padé approximation $q$-trigonometric functions

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