On solutions of two categories of \(q\)-shift equations in two dimensional complex field. (English) Zbl 07854811
Anal. Math. Phys. 14, No. 3, Paper No. 54, 15 p. (2024); correction ibid. 14, No. 4, Paper No. 80, 1 p. (2024).
Summary: In this paper, we mainly focus on finding transcendental entire solutions of binomial and trinomial equations generated by \(q\)-shift operator in \(\mathbb{C}^2\), to extend the results of
H. Li and H. Xu [Axioms 126, Paper No. 10, 19 p. (2021; doi:10.3390/axioms10020126)] and X. M. Zheng and H. Y. Xu [Anal. Math. 48, No. 1, 199–226 (2022; Zbl 1499.30292)]
completely in a different direction in terms of their \(q\)-shift counterpart. We have observed notable differences in the solutions derived from \(q\)-shift equations, including two different variants of \(q\)-difference equations, compared to the solutions obtained from the corresponding \(c\)-shift equations in \(\mathbb{C}^2\). Our findings have been supported with several examples that illustrate these differences. Additionally, the introduction of two new lemmas not explored in existing literature further adds depth to the mathematical tools available for addressing such problems.
MSC:
| 39B32 | Functional equations for complex functions |
| 39A45 | Difference equations in the complex domain |
| 39A70 | Difference operators |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 30D20 | Entire functions of one complex variable (general theory) |
| 30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |
Citations:
Zbl 1499.30292References:
| [1] | Baker, IN, On a class of meromorphic functions, Proc. Am. Math. Soc., 17, 819-822, 1966 · Zbl 0161.35203 · doi:10.1090/S0002-9939-1966-0197732-X |
| [2] | Gross, F., On the equation \(f^n+g^n=1\), Bull. Am. Math. Soc., 72, 86-88, 1966 · Zbl 0131.13603 · doi:10.1090/S0002-9904-1966-11429-5 |
| [3] | Gross, F., On the functional equation \(f^n+g^n=h^n\), Am. Math. Mon., 73, 10, 1093-1096, 1966 · Zbl 0154.40104 · doi:10.2307/2314644 |
| [4] | Gross, F., On the equation \(f^n+g^n=1\), II, Bull. Am. Math. Soc., 74, 4, 647-648, 1968 · Zbl 0159.20201 · doi:10.1090/S0002-9904-1968-11975-5 |
| [5] | Gross, F., Factorization of meromorphic functions, 1972, Naval Research Laboratory, Washington, DC: Mathematics Research Center, Naval Research Laboratory, Washington, DC · Zbl 0266.30006 |
| [6] | Hayman, WK, Meromorphic functions, 1964, Oxford: Clarendon Press, Oxford · Zbl 0115.06203 |
| [7] | Halburd, RG; Korhonen, RJ, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., 314, 477-487, 2006 · Zbl 1085.30026 · doi:10.1016/j.jmaa.2005.04.010 |
| [8] | Liu, K.; Cao, TB; Cao, HZ, Entire solutions of Fermat-type differential difference equations, Arch. Math., 99, 147-155, 2012 · Zbl 1270.34170 · doi:10.1007/s00013-012-0408-9 |
| [9] | Liu, K.; Yang, L., On entire solutions of some differential-difference equations, Comput. Methods Funct. Theory., 13, 433-447, 2013 · Zbl 1314.39022 · doi:10.1007/s40315-013-0030-2 |
| [10] | Liu, K.; Cao, TB, Entire solutions of Fermat-type \(q\)-difference differential equations, Electronic J. Diff. Equa., 2013, 59, 1-10, 2013 · Zbl 1287.39006 |
| [11] | Liu, K.; Yang, LZ, A note on meromorphic solutions of Fermat-types equations, An. Stiint. Univ. Al. I. Cuza Lasi Mat. (N. S.), 1, 317-325, 2016 · Zbl 1389.39038 |
| [12] | Liu, K.; Laine, I.; Yang, LZ, Complex delay-differential equations, 2021, Berlin: De Gruyter, Berlin · Zbl 1486.34001 · doi:10.1515/9783110560565 |
| [13] | Li, H.; Xu, H., Solutions for several quadratic trinomial difference equations and partial differential difference equations in \({\mathbb{C} }^2 \), Axioms, 126, 10, 1-19, 2021 |
| [14] | Luo, J.; Xu, YH; Hu, F., Entire solutions for several general quadratic trinomial differential difference equations, Open Math., 19, 1018-1028, 2021 · Zbl 1489.39022 · doi:10.1515/math-2021-0080 |
| [15] | Montel, P., Lecons sur les families normales de fonctions analytiqes et leurs applications, Collect, 135-136, 1927, Paris: Borel, Paris · JFM 53.0303.02 |
| [16] | Saleeby, EG, On complex analytic solutions of certain trinomial functional and partial differential equations, Aequationes Math., 85, 553-562, 2013 · Zbl 1267.30098 · doi:10.1007/s00010-012-0154-x |
| [17] | Stoll, W., Holomorphic functions of finite order in several complex variables, 1974, Providence: American Mathematical Society, Providence · Zbl 0292.32003 |
| [18] | Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. Math., 141, 443-551, 1995 · Zbl 0823.11029 · doi:10.2307/2118559 |
| [19] | Xu, L.; Cao, TB, Solutions of complex Fermat-type partial differential-difference equations, Mediterr. J. Math., 15, 227, 1-11, 2018 · Zbl 1403.39014 |
| [20] | Xu, L.; Cao, TB, Correction to: solutions of complex Fermat-type partial differential-difference equations, Mediterr. J. Math., 17, 1-4, 2020 · Zbl 1431.39008 · doi:10.1007/s00009-019-1438-3 |
| [21] | Yi, HX; Yang, CC, Uniqueness theory of meromorphic functions (2003), 1995, Dordrecht, Beijing: Kluwer Academic Publishers, Science Press, Dordrecht, Beijing |
| [22] | Zheng, XM; Xu, HY, Entire solution for some Fermat type functional equations concerning difference and partial differential in \({\mathbb{C} }^2 \), Anal. Math., 48, 1, 199-226, 2022 · Zbl 1499.30292 · doi:10.1007/s10476-021-0113-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
