Orthogonality of a new family of $q$-Sobolev type polynomials

In this work, we introduce and construct specific $q$-polynomials that are desired from the well-established families of $q$-orthogonal polynomials, namely little $q$-Jacobi polynomials and $q$-Laguerre polynomials, respectively. We examine these newly constructed $q$-polynomials and observe that they possess integral representations of little $q$-Jacobi polynomials and $q$-Laguerre polynomials. These polynomials solve a third-order $q$-difference equation and display an unconventional four-term recurrence relation. This unique recurrence relation makes us categorize them as $q$-Sobolev-type orthogonal polynomials. This motivation leads to defining the general Sobolev-type orthogonality for $q$-polynomials. Special cases of these polynomials are also explored and discussed. Furthermore, we delve into the behavior of these $q$-orthogonal polynomials of Sobolev type as the parameters approach $1$. We also examine their zeros and interlacing properties.