Moments of combinatorial and Catalan numbers

In this paper we obtain the moments {Φm}_m?0 defined by

     

Continued fractions,the group GL(2, ℤ), and Pisot numbers

The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.     

Approximation of inverse moments of discrete distributions

For a wide class of discrete distributions, we derive a representation of the inverse (negative) moments through the Stirling numbers of the first kind and inverse factorial moments. We specialize the results for the Poisson, binomial, hypergeometric and negative binomial distributions.     

Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Relationships for moments of order statistics from the right-truncated generalized half logistic distribution

In this paper, we establish several recurrence relations satisfied by the single and the product moments for order statistics from the right-truncated generalized half logistic distribution. These relationships may be used in a simple recursive manner in order to compute the single and the product moments of all order statistics for all sample sizes and for any choice of the truncation parameter P. These generalize the corresponding results for the generalized half logistic distribution derived recently by Balakrishnan and Sandhu (1995, J. Statist. Comput. Simulation, 52, 385–398).Earlier went by the name R. A. Sandhu.     

On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers

Let A_n=Circ(F₁,F₂,…,F_n) and B_n=Circ(L₁,L₂,…,L_n) be circulant matrices, where F_n is the Fibonacci number and L_n is the Lucas number. We prove that A_n is invertible for n > 2, and B_n is invertible for any positive integer n. Afterwards, the values of the determinants of matrices A_n and B_n can be expressed by utilizing only the Fibonacci and Lucas numbers. In addition, the inverses of matrices A_n and B_n are derived.     

Cauchy, Ferrers-Jackson and Chebyshev polynomials and identities for the powers of elements of some conjugate recurrence sequences

In this paper some decompositions of Cauchy polynomials, Ferrers-Jackson polynomials and polynomials of the form x ²ⁿ + y ²ⁿ , n ∈ ℕ, are studied. These decompositions are used to generate the identities for powers of Fibonacci and Lucas numbers as well as for powers of the so called conjugate recurrence sequences. Also, some new identities for Chebyshev polynomials of the first kind are presented here.     

高阶Euler数的推广及其应用

给出了高阶Euler数的一种Apostol型(看T.M.Apostol,[Pacific J.Math.,1(1951),161～167])推广,我们称之为高阶Apostol-Euler数,然后推导出它的几个递推公式并给出了它们的一些特殊情况和应用,从而得到了相应的高阶Euler数和经典Euler数的新公式.     

Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.      

On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers

In this paper, we have found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = C_r(F₀, F₁, …, F_n−1), B = C_r(L₀, L₁, …, L_n−1), and we have obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B matrices.     

Laws of large numbers for polynomial hypergroups and some applications

There are introduced moments on polynomial hypergroups. These moments are used to prove strong laws of large number (SSLLNs) for random walks on the nonnegative integers that are homogeneous with respect to a polynomial hypergroup where SLLNs of different kind appear for polynomial hypergroups thth different properties. Furthermore, we discuss polynomial hypergroups that are associated with some discrete semigroups in a canonical way, and, using SLLNs for polynomial hypergroups, we get SLLNs for isotropic random walks on some discrete semigroups.     

Powers of the Catalan Generating Function and Lagrange''s 1770 Trinomial Equation Series

The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.     

A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

We give a formula expressing Bernoulli numbers of the second kind as 2-adically convergent sums of traces of algebraic integers. We use this formula to prove and explain the formulas and conjectures of Adelberg concerning the initial 2-adic digits of these numbers. We also give analogous results for the Nörlund numbers.     

A generalization of the formula for the triangular number of the sum and product of natural numbers

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of r natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.     

An asymptotic formula for the moments of the Minkowski question mark function in the interval [0, 1]

In this paper, we prove an asymptotic formula for the moments of the Minkowski question mark function, which describes the distribution of rationals in the Farey tree. The main idea is to demonstrate that a certain variation of a Laplace method is applicable in this problem, and hence the task reduces to a number of technical calculations. Dedicated to Antanas Laurinčikas on the occasion of his 60th birthday     

Application of quadrature method of moments to the polydispersed droplet spectrum in transonic steam flows with primary and secondary nucleation

This paper investigates the effectiveness of the Quadrature Method of Moments (QMOM) in representing droplet size distributions present in low-pressure steam turbine stages. In particular, distributions that result during transonic flow with sufficient supercooling and high expansion rate for primary and secondary nucleation to occur along the flow path. It is shown that the discretization of the droplet size distribution inherent in QMOM is robust for representing a polydispersed distribution with sizes several orders of magnitude apart, and originating from separate condensation phenomena. Inclusion of the QMOM method would thus provide a significant improvement in condensation models currently in use, which generally rely on a monodispersed representation of the droplet distribution.     

An annulus for the zeros of polynomials

In this note we obtain a ring shaped region containing all the zeros of a polynomial involving binomial coefficients and Fibonacci's numbers.     

Randomly sampled Riemann sums and complete convergence in the law of large numbers for a case without identical distribution

Let the points be independently and uniformly randomly chosen in the intervals , where . We show that for a finite-valued measurable function on , the randomly sampled Riemann sums converge almost surely to a finite number as if and only if , in which case the limit must agree with the Lebesgue integral. One direction of the proof uses Bikelis' (1966) non-uniform estimate of the rate of convergence in the central limit theorem. We also generalize the notion of sums of i.i.d. random variables, subsuming the randomly sampled Riemann sums above, and we show that a result of Hsu, Robbins and Erd\H{o}s (1947, 1949) on complete convergence in the law of large numbers continues to hold. In the Appendix, we note that a theorem due to Baum and Katz (1965) on the rate of convergence in the law of large numbers also generalizes to our case.