Entire solutions of a certain type differential-difference equation and differential-difference analogue of Bruck conjecture
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911142916390912 |
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| author | Xu, Junfeng Majumder, Sujoy Pramanik, Debabrata |
| author_facet | Xu, Junfeng Majumder, Sujoy Pramanik, Debabrata |
| contents | In the paper, we find out the precise form of the finite order entire solutions of the following differential-difference equation \[f^{(k)}(z)=\sideset{}{^n_{j=0}}{\sum} a_j f(z+jc),\] where $a_0, a_1,\ldots,a_n(\neq 0)\in\mathbb{C}$. Also in the paper we study the differential-difference analogue of Brück conjecture and derive a uniqueness result of finite order entire function $f(z)$ having a Borel exceptional small function of $f(z)$, when $f^{(k)}(z)$ and $\sideset{}{^n_{j=0}}{\sum} a_j f(z+jc)$ share a small function of $f(z)$. The obtained results, significantly generalize and improve the results due to Liu and Dong (Some results related to complex differential-difference equations of certain types, Bull. Korean Math. Soc., 51 (5) (2014), 1453-1467). Some examples are given to ensure the necessity of the condition (s) of our main results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entire solutions of a certain type differential-difference equation and differential-difference analogue of Bruck conjecture Xu, Junfeng Majumder, Sujoy Pramanik, Debabrata Complex Variables 30D35, 39B32 and 34M10 In the paper, we find out the precise form of the finite order entire solutions of the following differential-difference equation \[f^{(k)}(z)=\sideset{}{^n_{j=0}}{\sum} a_j f(z+jc),\] where $a_0, a_1,\ldots,a_n(\neq 0)\in\mathbb{C}$. Also in the paper we study the differential-difference analogue of Brück conjecture and derive a uniqueness result of finite order entire function $f(z)$ having a Borel exceptional small function of $f(z)$, when $f^{(k)}(z)$ and $\sideset{}{^n_{j=0}}{\sum} a_j f(z+jc)$ share a small function of $f(z)$. The obtained results, significantly generalize and improve the results due to Liu and Dong (Some results related to complex differential-difference equations of certain types, Bull. Korean Math. Soc., 51 (5) (2014), 1453-1467). Some examples are given to ensure the necessity of the condition (s) of our main results. |
| title | Entire solutions of a certain type differential-difference equation and differential-difference analogue of Bruck conjecture |
| topic | Complex Variables 30D35, 39B32 and 34M10 |
| url | https://arxiv.org/abs/2509.06783 |