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The main aim of this article is to investigate some interesting symmetric identities for the Dirichlet-type multiple (p , q) -L function. We use this function to examine the symmetry of the generalized higher-order (p , q) -Euler polynomials related to χ. First, the generalized higher-order (p , q) -Euler numbers and polynomials related to χ are defined. We also give a few new symmetric properties for the Dirichlet-type multiple (p , q) -L-function and generalized higher-order (p , q) -Euler polynomials related to χ. [ABSTRACT FROM AUTHOR]

We get the 3-variable degenerate Hermite Kampé de Fériet polynomials and get symmetric identities for 3-variable degenerate Hermite Kampé de Fériet polynomials. We make differential equations coming from the generating functions of degenerate Hermite Kampé de Fériet polynomials to get some identities for 3-variable degenerate Hermite Kampé de Fériet polynomials,. Finally, we study the structure and symmetry of pattern about the zeros of the 3-variable degenerate Hermite Kampé de Fériet equations. [ABSTRACT FROM AUTHOR]

A family F is an intersecting family if any two members have a nonempty intersection. Erdős, Ko, and Rado showed that | F | ≤ n − 1 k − 1 holds for a k-uniform intersecting family F of subsets of [ n ] . The Erdős-Ko-Rado theorem for non-uniform intersecting families of subsets of [ n ] of size at most k can be easily proved by applying the above result to each uniform subfamily of a given family. It establishes that | F | ≤ n − 1 k − 1 + n − 1 k − 2 + ⋯ + n − 1 0 holds for non-uniform intersecting families of subsets of [ n ] of size at most k. In this paper, we prove that the same upper bound of the Erdős-Ko-Rado Theorem for k-uniform intersecting families of subsets of [ n ] holds also in the non-uniform family of subsets of [ n ] of size at least k and at most n − k with one more additional intersection condition. Our proof is based on the method of linearly independent polynomials. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate some interesting symmetric identities for Carlitz-type higher-order degenerate (p , q) -Euler numbers, and polynomials. At first, the Carlitz-type higher-order degenerate (p , q) -Euler numbers and polynomials are defined. We give few new symmetric identities for Carlitz-type higher-order degenerate (p , q) -Euler numbers and polynomials. [ABSTRACT FROM AUTHOR]

In this paper we define the degenerate Carlitz-type (p , q) -Euler polynomials by generalizing the degenerate Euler numbers and polynomials, degenerate Carlitz-type q-Euler numbers and polynomials. We also give some theorems and exact formulas, which have a connection to degenerate Carlitz-type (p , q) -Euler numbers and polynomials. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to find some interesting symmetric identities for the (p , q) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple (p , q) -Hurwitz-Euler eta function by generalizing the Carlitz's form (p , q) -Euler numbers and polynomials. We find some formulas and properties involved in Carlitz's form (p , q) -Euler numbers and polynomials with higher order. We find new symmetric identities for multiple (p , q) -Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz's form (p , q) -Euler numbers and polynomials with higher order by using symmetry about multiple (p , q) -Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz's form (p , q) -Euler numbers and polynomials with higher order. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to represent sums of finite products of Legendre and Laguerre polynomials in terms of several orthogonal polynomials. Indeed, by explicit computations we express each of them as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials, some of which involve terminating hypergeometric functions 1 F 1 and 2 F 1 . [ABSTRACT FROM AUTHOR]