In a similar manner as in the papers  and , where explicit algorithms for finding the differential equations satisfied by holonomic functions were given, in this paper we deal with the space of the q-holonomic functions which are the solutions of linear q-differential equations with polynomial coefficients. The sum, product and the composition with… (More)
We construct the sequence of orthogonal polynomials with respect to an inner product defined in the sense of q-integration over several intervals in the complex plane. For such introduced polynomials we prove that all zeros lie in the smallest convex hull over the intervals in the complex plane. The results are stated precisely in some special cases, as… (More)
Based on the fractional q–integral with the parametric lower limit of integration, we define fractional q–derivative of Riemann–Liouville and Ca-puto type. The properties are studied separately as well as relations between them. Also, we discuss properties of compositions of these operators.
We construct q-Taylor formula for the functions of several variables and develop some new methods for solving equations and systems of equations. They are much easier for application than the well-known ones. We introduce some values for measuring their accuracy, such as (r; q)-order of convergence. We made some analogue of known methods, such as q-Newton… (More)
Starting from q-Taylor formula for the functions of several variables and mean value theorems in q-calculus which we prove by ourselves, we develop a new methods for solving the systems of equations. We will prove its convergence and we will give an estimation of the error.
In this paper 1 , we discuss the properties of q-polynomials which are hold in any way, especially difference-differential equation and similar relations. Also, the distribution of zeros is studied. At last, we illustrate all by a few examples and make some conjectures.