AI 中文总结
研究截断Appell级数和截断q - Appell级数的同余性质，利用组合恒等式和‘创造性显微镜’方法，给出截断Appell级数\(F_1\)和\(F_2\)一些同余式的q - 类似，建立与\(F_4\)相关新q - 超同余式并猜想。
AI 中文摘要
近年来，Appell级数和截断Appell级数在学者中引起了极大的研究兴趣。在本文中，我们研究截断Appell级数和截断q - Appell级数的同余性质。通过使用各种组合恒等式和Guo与Zudilin引入的‘创造性显微镜’方法，我们为截断Appell级数\(F_1\)和\(F_2\)的几个已知同余式提供了q - 类似。同时，我们建立了一个与截断Appell级数\(F_4\)相关的新q - 超同余式并提出了一个相关猜想，这为Apagodu和Zeilberger的两个猜想提供了q - 类似。
英文摘要
In recent years, both Appell series and truncated Appell series have garnered significant research interest among scholars. In this paper, we investigate the congruence properties of truncated Appell series and truncated $q$-Appell series. By employing various combinatorial identities and the `creative microscoping' method introduced by Guo and Zudilin, we provide $q$-analogues for several known congruences of truncated Appell series $F_1$ and $F_2$. Meanwhile, we establish a new $q$-supercongruence related to the truncated Appell series $F_4$ and propose a related conjecture, which provides a $q$-analogue for two conjectures by Apagodu and Zeilberger.
