Sladjana D. Marinković

The inequalities for some types of q-integrals

Using the restriction of the q-integral over [a,b] to a finite sum and q-integral of Riemann-type, we establish new integral inequalities of q-Chebyshev type, q-Gruss type, q-Hermite-Hadamard type and Cauchy-Buniakowsky type. Some inequalities which include the boundaries of functions are also indicated.

The deformed and modified Mittag-Leffler polynomials

The starting point of this paper are the Mittag-Leffler polynomials investigated in details by Bateman (1940) in [4]. Based on generalized integer powers of real numbers and deformed exponential functions, we introduce deformed Mittag-Leffler polynomials defined by the appropriate generating function. We investigate their recurrence relations, hypergeometric representation and orthogonality. Since they have all zeros on the imaginary axes, we also consider real polynomials with real zeros associated to them.

The deformed exponential functions of two variables in the context of various statistical mechanics

In the recent development in various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In last two decades, the Tsallis and Kaniadakis versions have found a lot of applications. In this paper, we consider the deformations which have two purposes. First, we introduce them like beginning of a more general mathematical approach where the Tsallis and Kaniadakis exponential functions are the special cases. Then, we wish to pay attention to the mathematical community that they have a lot of interesting properties from mathematical point of view and possibilities in applications. Really, we will show the differential and difference properties of our deformations which are important for the formation and explanation of continuous and discrete models of numerous phenomena.

On q-Newton-Kantorovich method for solving systems of equations

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.

Properties of q-holonomic functions

In a similar manner as in the papers by W. Koepf, D. Schmersau, Spaces of functions satisfying simple differential equations, Konrad-Zuse-Zentrum Berlin (ZIB), Technical Report TR 94-2 (1994) and Salvy, B., Zimmermann, P., GFUN: A package for the manipulation of generating and holonomic functions in one variable, ACM Transactions on Mathematical Software, (1994), pp. 163–177, 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 power functions of q-holonomic functions are also q-holonomic and the resulting q-differential equations can be computed algorithmically.

On a Class of Almost Orthogonal Polynomials

In this paper, we define a new class of almost orthogonal polynomials which can be used successfully for modelling of electronic systems which generate orthonormal basis. Especially, for the classical weight function, they can be considered like a generalization of the classical orthogonal polynomials (Legendre, Laguerre, Hermite, ...). They are very suitable for analysis and synthesis of imperfect technical systems which are projected to generate orthogonal polynomials, but in the reality generate almost orthogonal polynomials.

Functions Induced by Iterated Deformed Laguerre Derivative: Analytical and Operational Approach

The one-parameter deformed exponential function was introduced as a frame that enchases a few known functions of this type. Such deformation requires the corresponding deformed operations (addition and subtraction) and deformed operators (derivative and antiderivative). In this paper, we will demonstrate this theory in researching of some functions defined by iterated deformed Laguerre operator. We study their properties, such as representation, orthogonality, generating function, differential and difference equation, and addition and summation formulas. Also, we consider these functions by the operational method.

The Almost Orthogonal Polynomial Sequences

Since the orthogonal polynomials have a lot of applications in mathematics (theory of special functions, spectral operators, probability, analytic numbers and approximations) and other sciences (physics, mechanics, electrotechnics), it appeared numerous extensions of the concept of standard orthogonality. We will discuss the concept of almost orthogonality which is less known, but very useful in modeling of electronic systems which generate orthonormal basis. They are very suitable for analysis and synthesis of imperfect technical systems which are projected to generate orthogonal polynomials, but in the reality generate almost orthogonal polynomials. Like the most important application we will consider approximations of the functions.

The Laplace transform induced by the deformed exponential function of two variables

Abstract Based on the easy computation of the direct transform and its inversion, the Laplace transform was used as an effective method for solving differential and integral equations. Its various generalizations appeared in order to be used for treating some new problems. They were based on the generalizations and deformations of the kernel function and of the notion of integral. Here, we expose our generalization of the Laplace transform based on the so-called deformed exponential function of two variables. We point out on some of its properties which hold on in the same or similar manner as in the case of the classical Laplace transform. Relations to a generalized Mittag-Leffler function and to a kind of fractional Riemann-Liouville type integral and derivative are exhibited.

On a connection between formulas about q–gamma functions

In this short communication, we want to pay attention to a few wrong formulas which are unfortunately cited and used in a dozen papers afterwards. We prove that the provided relations and asymptotic expansion about the q-gamma function are not correct. This is illustrated by numerous concrete counterexamples. The error came from the wrong assumption about the existence of a parameter which does not depend on anything. Here, we apply a similar procedure and derive a correct formula for the q-gamma function.