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Nevanlinna Theory for Jackson Difference Operators and Entire Solutions of q-Difference Equations

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Abstract

This paper has two purposes. One is to establish a version of Nevanlinna theory based on the historic so-called Jackson difference operator \({D_q}f(z) = {{f(qz) - f(z)} \over {qz - z}}\) for meromorphic functions of zero order in the complex plane ℂ. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem, five-value theorem and Wiman-Valiron theorem in sense of Jackson q-difference operator, which keep consistent with the classical Nevanlinna theory. The other is, by using this theory, to investigate entire solutions of linear q-difference equations concerning Jackson difference operators. We will study the growth of entire solutions of linear Jackson q-difference equations \(D_q^kf(z) + A(z)f(z) = 0\) with meromorphic coefficient A, where \(D_q^k\) is Jackson k-th order difference operator, and then to estimate the logarithmic order of some q-special functions. Further, we show that the growth of order of all admissible meromorphic solutions f of a general linear nonhomogeneous q-difference equations \({A_n}(z)D_q^nf(z) + \cdots + {A_1}(z){D_q}f(z) + {A_0}(z)f(z) = F(z)\) should be positive if δ (0, f) > 0.

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Authors and Affiliations

  1. Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, P. R. China

    T. B. Cao

  2. Department of Mathematics, Beijing University of Posts and Telecommunications, Beijing, 100876, P. R. China

    H. X. Dai

  3. School of Mathematic Sciences, Fudan University, Shanghai, 200433, P. R. China

    J. Wang

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  1. T. B. Cao
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  2. H. X. Dai
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  3. J. Wang
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Correspondence to T. B. Cao.

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The first and second authors are supported by the National Natural Science Foundation of China (No. 11871260).

The third author is supported by the National Natural Science Foundation of China (No. 11771090).

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Cao, T.B., Dai, H.X. & Wang, J. Nevanlinna Theory for Jackson Difference Operators and Entire Solutions of q-Difference Equations. Anal Math 47, 529–557 (2021). https://doi.org/10.1007/s10476-021-0092-8

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  • DOI: https://doi.org/10.1007/s10476-021-0092-8

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