In this note we consider the generalized Fibonacci and Lucas sequences given by: \[F_n=2aF_{n-1}+(b-a^2)F_{n-2}, F_0=0, F_1=1\]and \[L_n=2aL_{n-1}+(b-a^2)L_{n-2}, L_0=2, L_1=2a,\]where $a$ and $b$ are nonzero real numbers, and prove some of their properties. Also, we find their corresponding Q and R matrices as in the classical Fibonacci and Lucas sequences.

In this paper, we consider linear Diophantine fuzzification of substructures of a BE-algebra. In particular, we introduce the notion of linear Diophantine fuzzy $n$-fold weak subalgebra of a BE-algebra, study its properties, and relate it to the concept of $n$-fold weak subalgebra.

This paper aims to introduce two new concepts in mod-ule M over a ring R. The first one is called the e∗-lifting module, a generalization of the lifting module. The second one is called the F I- e∗-lifting modules, a proper inference of lifting (e∗-lifting) modules. Then we give some characterizations and properties of these concepts.

In this paper, we introduce the notion of tri-quasi ideals of ordered semirings which is a generalization of ideals, bi-ideals, quasi-ideals and interior ideals of ordered semirings and investigate the properties of tri-quasi ideals of ordered semirings. In addition, we study minimal tri-quasi ideals of ordered semirings and tri-quasi simple ordered semirings. Moreover, we introduce the concepts of fuzzy tri-quasi ideals, minimal fuzzy tri-quasi ideals of ordered semirings and fuzzy tri-quasi simple ordered semirings. The relationship between tri-quasi ideals and fuzzy tri-quasi ideals of ordered semirings is provided.

In this paper, we introduce the quasi-normality set, denoted by $ E_A $, which is a strong extension of the normality set, denoted by $ N_A$ for any operator $A$ in the Banach algebra $\mathcal{B(H)}$. Furthermore, we show some interesting properties and remarkable results with the Unilateral Weighted shift and Volterra operators. Finally, we show that a quasi-normality set is no invariant and has an invariant subspace for some transpose linear operators.

In this short note we prove a generalization of the binomial theorem within the context of enumerating induced weak compositions for a given fixed composition of an integer $n$. We then use our results to prove generalizations of well known binomial coefficient identities.

In this paper we give a new definition of fractional analytic functions which proved to be more useful and it will have more powerful and impact on the theory than other definitions. We use the new definition to develop theory of fractional analytic functions which includes fractional Cauchy-Riemann equations and fractional Cauchy integral formula.

We introduce the notion of a combinatorial game defined on a finite hyper dice backgammon (which is a generalization of the standard backgammon) by omitting the dice from it and call it Diceless or No Chance hyper dice backgammon. Instead of a random element, players choose their own numbers under certain constraints in full view of the opponent. The idea for this new game is simple and converts a non-combinatorial game (a game of chance) into an extended Partizan Combinatorial Game.

In this work, we present the concept of interval valued fuzzy bi-quasi ideals. We investigate the properties of an interval valued fuzzy bi-quasi ideal in semigroups. We characterize a regular semigroup in terms of an interval valued fuzzy bi-quasi ideal.

We will use the center-of-mass formula to find an upper bound for the Fibonacci and Zeckendorf representations of a positive integer by assuming that each Fibonacci number as representing the location of a physical point where we place a mass of size one unit of mass. We also find another upper bound for the Fibonacci and Zeckendorf representations of a positive integer by using an elementary arithmetical argument. Zeckendorf's theorem states that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. A sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation. Besides some comparative cases of two Fibonacci numbers, we will also define the Fibonacci representation and the center of mass of a positive integer by the uniqueness property of the Zeckendorf representation. Finally, we end the paper with a question related to the Hilbert's tenth problem and the center-of-mass technique.

Orthogonal polynomials are useful in many applications. Typically one can construct orthogonal polynomials on the interval $[a,b]$ through the Gram-Schmidt process. In this note we present a new way of constructing orthogonal polynomials on the interval $[a,b]$.

It is well known that if F is a finite field then F∗, the set of nonzero elements of F, is a cyclic group. In this paper we will assume F = Fp (the finite field with p elements, p a prime) and Fp2 is a quadratic extension of Fp. In this case, the groups F∗p and F∗ p2 have orders p −1 and p2 −1 respectively. We will provide necessary and sufficient conditions for an element u ∈ F∗ p2 to be a generator. Specifically, we will prove u is a generator of F∗ p2 if and only if N(u) generates F∗p and u2 N(u) generates Ker N, where N : F∗ p2 →F∗p denotes the norm map