Predrag M. Rajković

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 Hankel transform of the sum of consecutive generalized Catalan numbers

We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.

Influence of transverse shear on stochastic instability of viscoelastic beam

Abstract The stochastic instability problem associated with an axially loaded Timochenko beam made of viscoelastic material is formulated. The beam is treated as Voigt–Kelvin body compressed by time-dependent deterministic and stochastic forces. By using the direct Liapunov method, bounds of the almost sure instability of beams as a function of retardation time, variance of the stochastic force, mode number, section shape factor and intensity of the deterministic component of axial loading, are obtained. Calculations are performed for the Gaussian process with a zero mean and variance σ2 as well as for harmonic process with an amplitude A.

Pulse Energy Probability Density Functions for Long-Haul Optical Fiber Transmission Systems by Using Instantons and Edgeworth Expansion

In this work, we use a new approach to model pulse energy in long-haul optical fiber transmission systems. Existing approaches for obtaining probability density functions (pdfs) rely on numerical simulations or analytical approximations. Numerical simulations make far tails of the pdfs difficult to obtain, while analytical approximations are often inaccurate, as they neglect nonlinear interaction between pulses and noise. Our approach combines the instanton method from statistical mechanics to model far tails of the pdfs, with numerical simulations to refine the middle part of the pdfs. We combine the two methods by using an orthogonal polynomial expansion constructed specifically for this problem. We demonstrate the approach on an example of a specific submarine transmission system.

An Application of Sobolev Orthogonal Polynomials to the Computation of a Special Hankel Determinant

Many Hankel determinant computations arising in combinatorial analysis can be done using results from the theory of standard orthogonal polynomials. Here, we will emphasize special sequences which require the inclusion of discrete Sobolov orthogonality to find their closed form.

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.

The Hankel transform of generalized central trinomial coefficients and related sequences

In this paper, we explore the connection between the Hankel transform, Riordan arrays and orthogonal polynomials. For this purpose, we evaluate the Hankel transform of generalized trinomial coefficients, as a closed-form expression, using the method based on the orthogonal polynomials. Since the generalized trinomial coefficients are generalization of several integer sequences, obtained expression is also applicable in these cases. We also showed that the coefficient array of corresponding orthogonal polynomials can be represented in terms of Riordan arrays, which provides the LDLT decomposition of the Hankel matrix. Moreover, we consider the row sums of the inverse of coefficient array matrix and evaluate its Hankel transform.