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In this paper, we derive a family of ordinary differential equations from the generating function of the Laguerre polynomials. Then these differential equations are used in order to obtain some properties and new identities for those polynomials. [ABSTRACT FROM AUTHOR]

Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , ⋯ p − 1 } , where p is a prime and F = { F 1 , F 2 , … , F m } be a family of subsets of [ n ] such that | F i | (mod p ) ∈ K for all F i ∈ F and | F i ∩ F j | (mod p ) ∈ L for i ≠ j . In 1991 Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | F | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

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]

In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations. [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]

The main goal of this paper is to study some interesting identities for the multiple twisted (p , q) -L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted (p , q) -Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted (p , q) -Euler zeta function and multiple twisted (p , q) -L-function, which interpolate the Carlitz-type higher order twisted (p , q) -Euler numbers and Carlitz-type higher order twisted (p , q) -Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted (p , q) -Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted (p , q) -L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted (p , q) -Euler numbers and polynomials by using the symmetric property for the multiple twisted (p , q) -L-function. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating function of the (r , β) -Bell polynomials. We give explicit identities for the (r , β) -Bell polynomials. Finally, we find the zeros of the (r , β) -Bell equations with numerical experiments. [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]

Abstract: Let and be subsets of such that , where is a prime. Let be a family of subsets of with () for all and () for any . Every subset of can be represented by a binary code such that if and if . Alon–Babai–Suzuki proved in non-modular version that if for all , then . We generalize it in modular version. Alon–Babai–Suzuki also proved that the above bound still holds under and in modular version. Alon–Babai–Suzuki made a conjecture that if they drop one condition among and , then the above bound holds. But we prove the same bound under dropping the opposite condition . So we prove the same bound under only condition . This is a generalization of Frankl–Wilson theorem (Frankl and Wilson, 1981 ). [Copyright &y& Elsevier]