  We present a new approach to exponential functions on time scales and to
timescale analogues of ordinary differential 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\'e
approximants, we obtain Pad\'e-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.
