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The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences. [ABSTRACT FROM AUTHOR]

We introduce new types of distance Fibonacci numbers which are closely related with number decompositions. Using special decompositions of the number n we give a sequence of identities for them. Moreover, we give matrix generators for distance Fibonacci numbers and their direct formulas. [ABSTRACT FROM AUTHOR]

Thefirst main idea of this paper is to develop thematrix sequences that represent Padovan and Perrin numbers. Then, by taking into account matrix properties of these new matrix sequences, some behaviours of Padovan and Perrin numbers will be investigated. Moreover, some important relationships between Padovan and Perrin matrix sequences will be presented. [ABSTRACT FROM AUTHOR]

We give a new characterization of compact subsets of the fuzzy number space equipped with the level convergence topology. Based on this, it is shown that compactness is equivalent to sequential compactness on the fuzzy number space endowed with the level convergence topology. Our results imply that some previous compactness criteria are wrong. A counterexample also is given to validate this judgment. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give some properties of the modified q-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic p-adic q-integral on Zp, we derive new identities of q-Bernoulli numbers and polynomials with weight [ABSTRACT FROM AUTHOR]

We study the complete convergence and complete moment convergence for martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As a result, the Marcinkiewicz-Zygmund strong law of large numbers for martingale difference sequence is obtained. Our results generalize the corresponding ones of Stoica (2007, 2011). [ABSTRACT FROM AUTHOR]

Sufficient conditions on a sequence {ak} of nonnegative numbers are obtained that ensures f(z) = ... akzk is starlike of nonnegative order in the unit disk. A result of Vietoris on trigonometric sums is extended in this pursuit. Conditions for close to convexity and convexity in the direction of the imaginary axis are also established. These results are applied to investigate the starlikeness of functions involving the Gaussian hypergeometric functions. [ABSTRACT FROM AUTHOR]

Let {tn} ⊂ (0, 1) be such that tn ? 1 as n ? 8, let a and ß be two positive numbers such that a + ß = 1, and let f be a contraction. If T be a continuous asymptotically pseudocontractive selfmapping of a nonempty bounded closed convex subset K of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence {tn}, we show the existence of a sequence {xn}n satisfying the relation xn = (1 - tn/kn)f(xn) + (tn/kn)Tnxn and prove that {xn} converges strongly to the fixed point of T,which solves some variational inequality provided T is uniformly asymptotically regular. As an application, if T be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subsetK of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by z0 ∈ K, zn+1 = (1 - tn/kn)f(zn) + (atn/kn)Tnzn + (ßtn/kn)zn converges strongly to the fixed point of T. [ABSTRACT FROM AUTHOR]

Let {Xni,i ≥ 1, n ≥ 1} be an array of rowwise asymptotically almost negatively associated (AANA, in short) random variables. The complete convergence for weighted sums of arrays of rowwise AANA random variables is studied, which complements and improves the corresponding result of Baek et al. (2008). As applications, the Baum and Katz type result for arrays of rowwise AANA random variables and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of AANA random variables are obtained. [ABSTRACT FROM AUTHOR]

We consider certain probability problems which are naturally related to integer partitions. We show that the corresponding probabilities are values of classical modular forms. Thanks to this connection, we then show that certain ratios of probabilities are specializations of the Rogers-Ramanujan and Ramanujan- Selberg- Gordon-Göllnitz continued fractions. One particular evaluation depends on a result from Ramanujan's famous first letter to Hardy. [ABSTRACT FROM AUTHOR]

We completely classify all polynomials of type (x2 + x -(Δ-1))/4 which are prime or 1 for a range of consecutive integers x ≥ 0, called Rabinowitsch polynomials, where Δ ≡ 1(mod4) with Δ > 1 square-free. This corrects, extends, and completes the results by Byeon and Stark (2002, 2003) via the use of an updated version of what Andrew Granville has dubbed the Rabinowitsch-Mollin-Williams Theorem—by Granville and Mollin (2000) and Mollin (1996). Furthermore, we verify conjectures of this author and pose more based on the new data. [ABSTRACT FROM AUTHOR]

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a generalmethod to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented. [ABSTRACT FROM AUTHOR]