Gessel–Stanton's inverse series and a system of q-polynomials

The work incorporates an extension of the Lagrange inversion formula for x/(1−x)b+1 due to I. Gessel, D. Stanton, Application of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc. 277 (1) (1983) 173–201. A proposed general q-inversion pair provides extension to Askey–Wilson polynomial and q-Racah polynomial. This general pair is used to introduce a general class of q-polynomials which includes and extends the polynomials of little q-Jacobi, q-Laguerre, etc. Some properties of this class namely, Companion matrix, q-difference equation, etc., are also obtained.