The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second-order q-difference operator that maps polynomials of degree n to polynomials of degree \(n+1\). It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil made out of two q-Heun operators. The special case of the q-Heun operator associated to the little q-Jacobi polynomials is also treated.

中文翻译：

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大q-Jacobi类型的q-Heun算子和q-Heun代数

定义了指数网格上大q-Jacobi类型的q-Heun运算符。该算子是最通用的二阶q差分算子，它将阶n的多项式映射到阶\（n + 1 \）的多项式。它是由q-Pochhammer或大q-Jacobi多项式构成的基的对角线，在q-Hahn代数的算子中是双线性的。描述了包括q-Heun算符作为生成器的该代数的扩展。证明双正交Pastro多项式可以满足广义特征值问题，或等效地位于由两个q-Heun算子组成的特殊线性铅笔的核中。还处理了与小q-Jacobi多项式相关的q-Heun运算符的特例。