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

Pages: 302 - 311, Issue Special, September 2011

 Abstract        Full Text (335.4K)              

Jan L. Cieśliński - Uniwersytet w Białymstoku, Wydział Fizyki, ul. Lipowa 41, 15-424 Białystok, Poland (email)

Abstract: 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:  Time scales, exponential function, trigonometric functions, hyperbolic functions, Cayley transformation, fi rst and second order dynamic equations, Padé approximation, exact discretization, $q$-exponential function, $q$-trigonometric functions
Mathematics Subject Classification:  Primary: 33B10, 26E70; Secondary: 34N05, 65L12

Received: August 2010;      Revised: March 2011;      Published: October 2011.