Search

This paper deals with the Mittag-Leffler polynomials (MLP) by extracting their essence which consists of real polynomials with fine properties. They are orthogonal on the real line instead of the imaginary axes for MLP. Beside recurrence relations and zeros, we will point to the closed form of its Fourier transform. The most important contribution consists of the new differential properties, especially the finite and infinite differential equation.

We consider the linear combinations of elements of two sequences: the first one a priory given nonnegative sequence and the second random sequence from the unit interval. We investigate the expected value of the smallest natural number such that the value of these linear combinations exceed a positive number. After very clear geometrical conclusions, we find the function which expresses the expected value. Here, we recognize a few known results like the special cases., Comment: 9 pages, 5 figures

The starting point of this paper are the Mittag-Leffler polynomials introduced by H. Bateman [1]. Based on generalized integer powers of real numbers and deformed exponential function, we introduce deformed Mittag-Leffler polynomials defined by appropriate generating function. We investigate their recurrence relations, differential properties and orthogonality. Since they have all zeros on imaginary axes, we also consider real polynomials with real zeros associated to them.

In the recent development in a various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In this paper, we consider the deformations which have two purposes: to have them like special cases, and, even more, to acquit their inauguration from mathematical point of view. Really, we will show interesting differential and difference properties of our deformations which are important in forming and explanation of continuous and discrete models of numerous phenomena.

Based on the fractional $q$-integral with the parametric lower limit of integration, we define fractional $q$-derivative of Riemann-Liouville and Caputo type. The properties are studied separately as well as relations between them. Also, we discuss properties of compositions of these operators.

We discuss the inequalities for $q$-integrals because of the fact that the inequalities can be very useful in the future mathematical research. Since $q$-integral of a function over an interval $[a,b]$ is defined by the difference of two infinite sums, there a lot of unexpected troubles in analyzing analogs of well-known integral inequalities. In this paper, we will signify to some directions to exceed this problem.