We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind JS(n+ k, n; z) by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting JS(n + k, n; z) = pk,0(n) + pk,1(n)z + · · ·+ pk,k(n)z, we show that (1− t)3k−i+1 ∑ n≥0 pk,i(n)t n is a polynomial in t with nonnegative integral coefficients and provide com...

For an arbitrary finite set S of natural numbers greater 1, we construct f ∈ Int(Z)= {g∈Q[x] | g(Z)⊆Z} such that S is the set of lengths of f , i.e., the set of all n such that f has a factorization as a product of n irreducibles in Int(Z). More generally, we can realize any finite multi-set of natural numbers greater 1 as the multi-set of lengths of the essentially different factorizations of ...

Abstract. We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind JS(n+ k, n; z) by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting JS(n + k, n; z) = pk,0(n) + pk,1(n)z + · · ·+ pk,k(n)z , we show that (1− t) ∑ n≥0 pk,i(n)t n is a polynomial in t with nonnegative integral coefficients and provid...

A $it{weighted~slant~Toep}$-$it{Hank}$ operator $L_{phi}^{beta}$ with symbol $phiin L^{infty}(beta)$ is an operator on $L^2(beta)$ whose representing matrix consists of all even (odd) columns from a weighted slant Hankel (slant weighted Toeplitz) matrix, $beta={beta_n}_{nin mathbb{Z}}$ be a sequence of positive numbers with $beta_0=1$. A matrix characterization for an operator to be $it{weighte...

In this paper we investigate the existence, boundedness and asymptotic behavior of positive solutions fuzzy difference equation \[z_{n+1}=\dfrac{Az_{n-1}}{1+z_{n-2}^{p}},~n\in\mathbb{N}_{0}\] where ( z n ) (zn) is a sequence numbers, A initial conditions ? j z?j = 0 , 1 2 (j=0,1,2) are numbers p integer.

A Z-number is very powerful in describing imperfect information, which fuzzy numbers are paired such that the partially reliable information properly processed. During a decision-making process, human beings always use natural language to describe their preferences, and decision usually imprecise reliable. The nature of Z-number, composed restriction reliability components, has made it tool for...

Quantum mechanics, at least at some instrumental level, can be viewed as a modification of the probability theory. We replace positive numbers (probabilities) with complex numbers z (probability amplitudes) such that the squares of their absolute values, |z|2, are interpreted as probabilities. The rules for combining amplitudes are very reminiscent of the rules for combining probabilities. z1 z2

We present the predictions for the photometric and emission line properties of galaxies present during the latter stages of reionization from z = 8 → 6. These preliminary predictions are made from cosmological hydrodynamic simulations that include star formation and feedback, but not the effects of radiative transfer. We find significant numbers of galaxies that have stellar masses exceeding 10...

Let Z be a finite set of double points in P 1 × P 1 and suppose further that X, the support of Z, is arithmetically Cohen-Macaulay (ACM). The scheme Z is rarely ACM. In this note we present an algorithm, which depends only upon a combinatorial description of X, for the bigraded Betti numbers of I Z , the defining ideal of Z.

We present a piecewise formula for the competition numbers of the complete tripartite graphs. For positive integers x, y and z where 2 ≤ x ≤ y ≤ z, the competition number of the complete tripartite graph Kx,y,z is yz − z − y − x + 3 whenever x 6= y and yz − 2y − z + 4 otherwise.