Basic Recurrence Relation Research Articles

Generating functions for vector partition functions and a basic recurrence relation

ABSTRACTWe define a generalized vector partition function and derive an identity for the generating series of such functions associated with solutions to basic recurrence relations of combinatorial analysis. As a consequence we obtain the generating function of the number of generalized lattice paths and a new version of the Chaundy-Bullard identity for the vector partition function.

A generalization of Hermite polynomials

In this paper, we introduce a simple set   ( ) p n H x , which is a generalized form of Hermite polynomial ( ) n H x . We establish series form, basic recurrence relations, the pure recurrence relation and Rodrigues type formula for the generalized polynomial. Mathematics Subject Classification: 33C45, 11B37, 05A15

An extension of Hermite polynomials

In this paper, we introduce an extended Hermite polynomial ( , ) ( ) a b n H x , we establish its different forms, generating functions, basic recurrence relations, the pure recurrence relation, the second order differential equation, Rodrigues formula and orthogonality of the extended polynomial. Mathematics Subject Classification: 33C45, 11B37, 05A15

Orthogonal polynomials—centroid of their zeroes

Let c (n,k) (k=1,...,n) the n zeroes of the monic orthogonal polynomials family P (n) (x). The centroid of these zeroes: controls globally the distribution of the zeroes, and it is relatively easy to obtain information on s (n) , like bounds, inequalities, parameters dependence, ..., from the links between s (n) , the coefficients of the expansion of P (n) (x), and the coefficients beta (n) , gamma (n) in the basic recurrence relation satisfied by P (n) (x). After a review of basic properties of the centroid on polynomials, this work gives some results on the centroid of a large class of orthogonal polynomials.

A Note on Order Statistics for Dependent Variates

Abstract Sathe and Dixit showed that a basic recurrence relation for cdf's of order statistics can be modified to hold under any dependence structure of the original variates. A greatly shortened proof is given in this note. Balakrishnan, Bendre, and Malik have used the relation to obtain a general expression for the cdf of an order statistic. This result is here linked to a classical formula in probability theory.

A Note on Moments of Order Statistics

Abstract In this article we show that two well-known identities among moments of order statistics follow directly from some basic recurrence relations satisfied by these moments. Based on this result, we claim that it is not meaningful to apply these identities for the purpose of checking the computations of these moments whenever these recurrence relations are used in the computational procedure.

Can the Velocity Autocorrelation Function Decay Exponentially?

For Hermitian many-body systems, admissible functions for the velocity autocorrelation function form a certain invariant of time. This property originates from a basic recurrence relation. The exponentially decaying function cannot form an invariant and therefore cannot be admissible.

Coefficient Identities for Powers of Taylor and Dirichlet Series

giving a way to calculate as many of the B's as desired. Formula (1.2), stated in tha way or in the form (1.3) or in various other forms, has been known a long time. Thus, in 1748 Euler gave (1.3) (obscurely expressed) in section 76 of his Introductio [4], [5]. Adams and Hippisley [1] gave the formula in the form (1.2) (their formula 6.361). Formula (1.3) is given as formula (0.314) in any of the several editions (Russian original, German, or English) of Ryshik and Gradstein [16], where, however, p is restricted to be a natural number. Thinking to remove this restriction of Ryshik and Gradstein, von Holdt [17] published still another derivation using properties of double sums to establish that (1.2) holds true for rational real p. His derivation avoids use of differentiation of the series in (1.1). We mention this because differentiation of formal power series affords a quick proof of (1.2) and has been used before. The basic recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. Thus Barrucand [2] found (1.2) again and made applications of it. Cappellucci [18] found it in the form (1.3) but attributed it to Hansted [19]. Hindenburg [12], in his treatise on the multinomial theorem (p. 291) gave (1.3) in a perfectly obscure notation. Many other references could be cited. Actually, the recurrence is implicit in still another class of formulas widespread in the literature, stemming from the early work of Hindenburg's student Rothe [15]. I myself have written a number of papers, e.g., [6]-[11] having to do with a formula of Rothe and its consequences for combinatorics, special functions and number theory. What we shall do here is to derive the formulas again and put them in a

On Asymptotic Normality in Stochastic Approximation

A new method, simpler than previous methods due to Chung (1954) and Sacks (1958), is used to prove Theorem 2.2 below, which implies in a simple way all known results on asymptotic normality in various cases of stochastic approximation. Two examples of application are concerned with Venter's (1967) extension of the RM method and Fabian's (1967) modification of the KW process. Previously, although there was no difficulty in adopting one or the other method, the proofs in various cases had to be done almost ab initio or skipped leaving a gap (see Venter (1967)). The new proof is similar to that of Chung except that the basic recurrence relation is used to obtain the asymptotic characteristic function rather than limits of all moments. We remark that Lemma 2.1, a simple corollary to Chung's lemma is used only to obtain condition (2.2.4) which is weaker than (2.2.3) if $\alpha = 1$ and which corresponds to the usual Lindeberg condition. Both conditions (2.2.3) and (2.2.4) are weaker than the corresponding condition (3.4) in Sacks (1958). In what follows $(\Omega, \mathscr{S}, P)$ will be a probability space, relations between and convergence of random variables, vectors, and matrices will be meant with probability one unless specified otherwise. We shall write $X_n \sim \mathscr{L}$ if $X_n$ is asymptotically $\mathscr{L}$-distributed and $X_n \sim Y_n$, for two sequences of random vectors, if for any $\mathscr{L}, X_n \sim \mathscr{L}$ if and only if $Y_n \sim \mathscr{L}$. The indicator function of a set $A$ will be denoted by $\chi A$, the expectation and conditional expectation by $E$ and $E_F$, respectively. $R^k$ is the $k$-dimensional Euclidean space the elements of which are considered to be column vectors, $R = R^1, R^{k\times k}$ is the space of all real $k \times k$ matrices. The symbols $\mathbf{R}, \mathbf{R}^k, \mathbf{R}^{k\times k}$, denote sets of all measurable transformations from $(\Omega, \mathscr{S})$ to $R, R^k R^{k\times k}$, respectively. The unit matrix in $R^{k\times k}$ is denoted by $I$ and $| |$ is the Euclidean norm. With $h_n$ a sequence of numbers, $o(h_n), O(h_n), o_u(h_n), O_u(h_n)$ denote sequences $g_n, G_n, q_n, Q_n$, say, of elements in one of the sets $\mathbf{R}, \mathbf{R}^k, \mathbf{R}^{k\times k}$ such that $h_n^{-1} g_n \rightarrow 0, | h_n^{-1} G_n|\leqq f$ for an $f \varepsilon \mathbf{R}$ and all $n, h_n^{-1} q_n \rightarrow O$ uniformly on a set of probability one, $| h_n^{-1} Q_n| \leqq K$ for a $K \varepsilon R$ and all $n$. In special cases $o(h_n)$ may be constant on $\Omega$ and considered as a sequence with elements in $R, R^k$ or $R^{k\times k}$. Similarly in other cases. For Chung's lemma, which will be frequently referred to, or used later without reference, see Fabian ((1967), Lemma 4.2); note that it holds with $\beta = 0$, too.