Finite logarithmic order solutions of linear $q$-difference equations
Bull. Korean Math. Soc. 2014 Vol. 51, No. 1, 83-98
https://doi.org/10.4134/BKMS.2014.51.1.83
Printed January 1, 2014
Zhi-Tao Wen
Department of Physics and Mathematics
Abstract : During the last decade, several papers have focused on linear $q$-difference equations of the form $$ \sum_{j=0}^na_j(z)f(q^jz)=a_{n+1}(z) $$ 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.
Keywords : logarithmic Borel exceptional value, logarithmic derivative, logarithmic exponent of convergence, logarithmic order, $q$-Casorati determinant, $q$-difference equation
MSC numbers : Primary 39A13; Secondary 30D35
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