Wiman-Valiron theorem for q-differences

Abstract

Abstract. Let q be any complex number other than 0 and 1. We first asymptotically express the logarithmic q-difference log f (qz) − log f (z) in terms of the logarithmic derivative f ′ /f for any meromorphic function f of order strictly less than 1/2. Then we show the assumption that the order strictly less than 1/2 is sharp. Finally, we prove a q-difference analogue of the Wiman-Valiron theorem for entire functions of order strictly less than 1/2.

Nevanlinna theory for Jackson difference operators and entire solutions of q-difference equations

“…By Wen and Ye's Wiman-Valiron theorem for q-difference [39], we obtain a Wiman-Valiron theorem for Jackson difference. Lemma 4.1.…”

A continuous transition from $\mathcal{E}$-sets to $R$-sets and beyond

“…A q-difference counterpart to Gundersen's pointwise estimates was discovered by Wen and Ye in [17]. For a K-version of it, all we need is to change the estimate (3.9) in [17] by the reasoning used in proving Theorem 4.1, and use the same constants d ν , and then [17,Lemma 3.4] yields the following analogue of (4.3):…”

“…Gundersen used R-sets and E-sets (but under different teminology) in finding sharp estimates for logarithmic derivatives of meromorphic functions [8]. Applying these findings, Chiang and Feng obtained pointwise estimates for logarithmic differences of meromorphic functions [3], while Wen and Ye estimated logarithmic q-differences of meromorphic functions [17]. Estimates in these directions have had numerous applications in the theories of complex differential equations and complex difference equations.…”

On the meromorphic solutions of generalized q-difference equations

“…By the use of recent new difference versions of Nevanlinna theory, many studies [1,5,[11][12][13][14][15][16][17][18][19][20] of the properties of meromorphic solutions of the difference equation…”