In this work, a thorough mathematical framework for incorporating Large Language Models (LLMs) into gamified systems is presented with an emphasis on improving task dynamics, user engagement, and reward systems. Personalized feedback, adaptive learning, and dynamic content creation are all made possible by integrating LLMs and are crucial for improving user engagement and system performance. A simulated environment tests the framework's adaptability and demonstrates its potential for real-world applications in various industries, including business, healthcare, and education. The findings demonstrate how LLMs can offer customized experiences that raise system effectiveness and user retention. This study also examines the difficulties this framework aims to solve, highlighting its importance in maximizing involvement and encouraging sustained behavioral change in a range of sectors.

翻译：本研究提出了一个将大型语言模型（LLMs）整合到游戏化系统中的严谨数学框架，重点在于优化任务动态、提升用户参与度以及改进奖励机制。通过集成LLMs，系统能够实现个性化反馈、自适应学习与动态内容生成，这些功能对于增强用户参与度和提升系统性能至关重要。我们在模拟环境中测试了该框架的适应性，并展示了其在商业、医疗保健及教育等多个行业实际应用的潜力。研究结果表明，LLMs能够提供定制化体验，从而提高系统效能与用户留存率。本文还探讨了该框架旨在解决的关键挑战，强调了其在最大化用户参与度及促进跨行业持续性行为改变方面的重要意义。