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

Jan L. Cieśliński

doi:10.3934/proc.2011.2011.302

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2011, Volume 2011: 302-311. Doi: 10.3934/proc.2011.2011.302 open access

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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

Received: July 31, 2010

Revised: February 28, 2011

Published: August 2017

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Abstract

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:

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

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

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