New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown.

中文翻译：

---

具有超指数权重函数的正交多项式：Ditkin-Prudnikov 问题的显式解

发现了具有超指数权重函数的正交多项式的新序列。特别是，我们给出了 Ditkin-Prudnikov 问题 (1966) 的明确解决方案。推导出了三项递推关系、显式表示、生成函数和罗德里格斯式公式。该方法基于所涉及的特殊函数的微分性质及其在梅林-巴恩斯和拉普拉斯积分方面的表示。引入了复合多项式正交性的概念。显示了这种正交性在发现多项式的新序列及其与相应的多个正交多项式系综的关系方面的相应优势。