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The main purpose of this paper is to construct new families of special numbers with their generating functions. These numbers are related to many well-known numbers, which are Bernoulli numbers, Fibonacci numbers, Lucas numbers, Stirling numbers of the second kind and central factorial numbers. Our other inspiration of this paper is related to the Golombek’s problem [15] “Aufgabe 1088. El. Math., 49 (1994), 126–127”. Our first numbers are not only related to the Golombek's problem, but also computation of the negative order Euler numbers. We compute a few values of the numbers which are given by tables. We give some applications in probability and statistics. That is, special values of mathematical expectation of the binomial distribution and the Bernstein polynomials give us the value of our numbers. Taking derivative of our generating functions, we give partial differential equations and also functional equations. By using these equations, we derive recurrence relations and some formulas of our numbers. Moreover, we come up with a conjecture with two open questions related to our new numbers. We give two algorithms for computation of our numbers. We also give some combinatorial applications, further remarks on our new numbers and their generating functions.

In current study, we focus on a mathematical concept called the k-hypergeometric polynomials. These polynomials are constructed using a mathematical tool called the Pochhammer k-symbol, as introduced by [R. Diaz, E. Pariguan, On hypergeometric functions and pochhammer k-symbol, Divulg. Mat. 15(2007), 179-192]. We develop several theorems related to these k-hypergeometric polynomials. Using these theorems, we derive two important functions: a multilinear generating function and a multilateral generating function for k-hypergeometric polynomials. These functions play a crucial role in our analysis. Furthermore, extend our research to explore the concept of the k-fractional secondary driver. This extension is based on the properties of k-hypergeometric polynomials and another mathematical entity known as the beta k-function. To make these connections, we utilize the Riemann-Liouville k-fractional process, as described by [G. Rahman, S. Nisar Mubeen, K. Sooppy, On generalized k-fractional derivative operator, AIMS Math. 5(2020), 1936-1945]. This has allowed us to establish some novel results, which are analogous to well-known mathematical transformations like the Mellin transformation. Additionally, we explore the relationships between our findings and other mathematical functions, such as hypergeometric and Appell’ k-functions. In the last section of our paper, we delve into the relationship between k-hypergeometric polynomials and two specific mathematical functions: We also provide an integral representation of k-hypergeometric polynomials. Overall, our research paper contributes to the understanding of k-hypergeometric polynomials and their connections to various mathematical functions and transformations.

It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, f(x), by the tangential affine function that passes through the point (E{X},f(E{X})), where E{X} is the expectation of the random variable X. While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to f, it turns out that when the function f is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than (E{X},f(E{X})). In this paper, we take advantage of this observation by optimizing the point of tangency with regard to the specific given expression in a variety of cases and thereby derive several families of inequalities, henceforth referred to as “Jensen-like” inequalities, which are new to the best knowledge of the author. The degree of tightness and the potential usefulness of these inequalities is demonstrated in several application examples related to information theory.

This paper introduces the concept of entropic value-at-risk (EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the value-at-risk (VaR) as well as the conditional value-at-risk (CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.

There are several general concepts that allow obtaining explicit formulas for the coefficients of generating functions in one variable by using their powers. One such concept is the application of compositae of generating functions. In previous studies, we have introduced a generalization for the compositae of multivariate generating functions and have defined basic operations on the compositae of bivariate generating functions. The use of these operations helps to obtain explicit formulas for compositae and coefficients of generating functions in two variables. In this paper, we expand these operations on compositae to the case of generating functions in three variables. In addition, we describe a way of applying compositae to obtain coefficients of rational generating functions in several variables. To confirm the effectiveness of using the proposed method, we present detailed examples of its application in obtaining explicit formulas for the coefficients of a generating function related to the Aztec diamond and a generating function related to the permutations with cycles.

The present paper deals with various recurrence relations, generating functions and series expansion formulas for two families of orthogonal polynomials in two variables, given Laguerre–Laguerre Koornwinder polynomials and Laguerre–Jacobi Koornwinder polynomials in the limit cases. Several families of bilinear and bilateral generating functions are derived. Furthermore, some special cases of the results presented in this study are indicated.

This paper investigates the behavior of asset prices in an endowment economy in which a representative agent with power utility consumes the dividends of multiple assets. The assets are Lucas trees; a collection of Lucas trees is a Lucas orchard. The model generates return correlations that vary endogenously, spiking at times of disaster. Since disasters spread across assets, the model generates large risk premia even for assets with stable cashflows. Very small assets may comove endogenously and hence earn positive risk premia even if their cashflows are independent of the rest of the economy. I provide conditions under which the variation in a small asset's price-dividend ratio can be attributed almost entirely to variation in its risk premium.

In this paper, our aim is to build generalized homogeneous q-difference equations for q-polynomials. We also consider their applications to generating functions and fractional q-integrals by using the perspective of q-difference equations. In addition, we also reveal relevant relations of various special cases of our main results involving some known results.

In the present paper, we study the information generating (IG) function and relative information generating (RIG) function measures associated with maximum and minimum ranked set sampling (RSS) schemes with unequal sizes. We also examine the IG measures for simple random sampling (SRS) and provide some comparison results between SRS and RSS procedures in terms of dispersive stochastic ordering. Finally, we discuss the RIG divergence measure between SRS and RSS frameworks.