Abstract: In this paper, systematic approaches to determining polynomial stress functions for quasicrystal plane problems were presented based on the generalized Lekhnitskii’s anisotropic elasticity theory. The approaches were applied to develop hybrid stress function (HSF) finite elements. Results show that for quasicrystal plane problems, an arbitrary nth-degree homogeneous polynomial encompasses a maximum of six independent polynomials, and the general expression of the polynomial stress function can be explicitly expressed. The obtained polynomials are used as analytical trial functions to construct a novel 8-node hybrid stress function (HSF) element. In comparison with traditional numerical methods, HSF demonstrates higher accuracy and superior performance.