From the literature it is known that orthogonal polynomials as the Jacobi polynomials can be expressed by hypergeometric series. In this paper, the authors derive several contiguous relations for terminating multivariate hypergeometric series. With these contiguous relations one can prove several recursion formulas of those series. This theoretical result allows to compute integrals over products of Jacobi polynomials in a very efficient recursive way. Moreover, the authors present an application to numerical analysis where it can be used in algorithms which compute the approximate solution of boundary value problem of partial differential equations by means of the finite elements method (FEM). With the aid of the contiguous relations, the approximate solution can be computed much faster than using numerical integration. A numerical example illustrates this effect.

中文翻译：

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Jacobi多项式的积分乘积的递归公式

从文献中知道，可以通过超几何级数来表示作为雅可比多项式的正交多项式。在本文中，作者得出了一些终止多元超几何级数的连续关系。利用这些连续关系，可以证明这些系列的几个递推公式。该理论结果允许以非常有效的递归方式计算Jacobi多项式乘积的积分。此外，作者提出了一种在数值分析中的应用，该方法可用于通过有限元方法（FEM）计算偏微分方程的边值问题的近似解的算法中。借助连续关系，可以比使用数值积分更快地计算出近似解。