A new non-Hermitian E2-quasi-exactly solvable model

•A previously unknown quasi-exactly solvable non-Hermitian model has been constructed.•The explicit eigenfunctions for the model have been constructed.•A double scaling limit leading to the Mathieu equation has been carried out.•The norms, Stieltjes measures and moment functionals are evaluated.

We construct a previously unknown E2E2-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model becomes Hermitian when one of its two parameters is fixed to a specific value. We analyze the double scaling limit of this model leading to the complex Mathieu equation. The norms, Stieltjes measures and moment functionals are evaluated for some concrete values of one of the two parameters.