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FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

  • Wen, Zhi-Tao (Taiyuan University of Technology Department of Mathematics, University of Eastern Finland Department of Physics and Mathematics)
  • Received : 2012.09.20
  • Published : 2014.01.31
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Abstract

During the last decade, several papers have focused on linear q-difference equations of the form ${\sum}^n_{j=0}a_j(z)f(q^jz)=a_{n+1}(z)$ with entire or meromorphic coefficients. A tool for studying these equations is a q-difference analogue of the lemma on the logarithmic derivative, valid for meromorphic functions of finite logarithmic order ${\rho}_{log}$. It is shown, under certain assumptions, that ${\rho}_{log}(f)$ = max${{\rho}_{log}(a_j)}$ + 1. Moreover, it is illustrated that a q-Casorati determinant plays a similar role in the theory of linear q-difference equations as a Wronskian determinant in the theory of linear differential equations. As a consequence of the main results, it follows that the q-gamma function and the q-exponential functions all have logarithmic order two.

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Cited by

  1. Growth of Meromorphic Solutions to Some Complex Functional Equations vol.16, pp.3, 2016, https://doi.org/10.1007/s40315-016-0157-z
  2. Growth of Meromorphic Solutions of Finite Logarithmic Order of Linear Difference Equations vol.54, pp.1, 2015, https://doi.org/10.1515/fascmath-2015-0001