The (p,q)-Analogues of Bi-Stirling Eulerian Polynomials via continued fractions

We establish Jacobi-type continued fraction expansions for (p,q)-analogues of the bi-Stirling Eulerian polynomials, which refine classical Eulerian statistics on permutations. These polynomials admit equivalent definitions in terms of descents or excedances. The parameters p and q encode refined permutation statistics, including (ldes,rasc) and (cros,nest). Our approach is based on weighted Motzkin path models arising from variants of the bijections of Françon–Viennot and Foata–Zeilberger. As applications, we obtain new results on total positivity and gamma-positivity for several families of enumerative polynomials. Our results unify and extend a number of recent works in the literature.