On certain generalized families of unified elliptic-type integrals pertaining to Euler integrals and generating functions

Elliptic-type integrals have their importance and potential in certain problems in radiation physics and nuclear technology. A number of earlier works on the subject contains several interesting unifications and generalizations of some significant families of elliptic-type integrals. The present paper is intended to obtain certain new theorems on generating functions. The results obtained in this paper are of manifold generality and basic in nature. Beside deriving various known and new elliptic-type integrals and their generalizations these theorems can be used to evaluate various Euler-type integrals involving a number of generating functions.      

任意随机序列关于非齐次马氏链的随机和的一类随机偏差定理

In this paper, the notion of limit random logarithmic likelihood ratio of stochastic sequence, as a measure of dissimilarity between the joint distribution on measure P and the Markov distribution on measure Q, is introduced. A class of random approximation theorems for arbitrary stochastic dominated sequence are obtained by using the tools of generating functions and the tailed-probability generating functions.     

C半群的概率逼近问题

借助Pettis积分、随机过程、矩生成函数及算子值数学期望,给出了一般形式的C半群概率逼近指数公式、生成定理及其收敛速度的估计式,也从另一个角度得出C半群概率表示的Vonorovskaya型渐近公式.     

An exposá of the mullin-rota theory of polynomials of binomial type

The basic theorems of the Mullin-Rota theory of polynomials of binomial type are rederived here by a combination of the classical theory of generating functions and Rota's operator theoretical methods. As a result a certain number of new identities are obtained.     

M阶非齐次马氏链随机转移概率调和平均的一类强极限定理

主要研究任意m阶非齐次马氏链的随机转移概率调和平均的a.s.收敛的强极限定理.在证明中采用了一种把网微分法与条件矩母函数相结合应用于马氏链强极限定理研究的新途径.作为推论,得到m阶非齐次马氏链的一个公平比的强极限定理,并将已有的结果加以推广.     

An exposá of the mullin-rota theory of polynomials of binomial type

The basic theorems of the Mullin-Rota theory of polynomials of binomial type are rederived here by a combination of the classical theory of generating functions and Rota's operator theoretical methods. As a result a certain number of new identities are obtained.     

m阶马氏链关于广义随机选择系统的极限定理

主要研究广义随机选择系统中的m阶非齐次马氏链随机转移概率调和平均的a.s.收敛的强极限定理.在证明中采用了一种把网微分法与条件矩母函数相结合应用于马氏链强极限定理研究的新途径.作为推论,得到m阶非齐次马氏链随机条件概率一个公平比的强极限定理,并将已有的结果加以推广.     

Discrete semi-stable distributions

The purpose of this paper is to introduce and study the concepts of discrete semi-stability and geometric semi-stability for distributions with support inZ ₊. We offer several properties, including characterizations, of discrete semi-stable distributions. We establish that these distributions posses the property of infinite divisibility and that their probability generating functions admit canonical representations that are analogous to those of their continuous counterparts. Properties of discrete geometric semi-stable distributions are deduced from the results obtained for discrete semi-stability. Several limit theorems are established and some examples are constructed.     

Groups of partial differential operators and the generalized Bessel functions

Considering the generalized Bessel functions satisfying a special ordinary differential equation of mth order, we derive some addition theorems and generating functions with the help of some algebra constructed for a group of first order partial differential operators grounding on the recurrence relations for these functions.     

子结构

用有别于格的方式,把群的子群,环的子环(理想)和BCK-代数的子代数(理想)等数学对象统一起来.引入子结构、子、生成子和既约子的概念,并借助于拓扑中的方法研究它们的性质.得到了子、生成子和既约子的性质及子的既约分解定理.把BCK-代数的理想分解定理推广到了众多的数学对象上.     

Permutations and words counted by consecutive patterns

Generating functions which count occurrences of consecutive sequences in a permutation or a word which matches a given pattern are studied by exploiting the combinatorics associated with symmetric functions. Our theorems take the generating function for the number of permutations which do not contain a certain pattern and give generating functions refining permutations by the both the total number of pattern matches and the number of non-overlapping pattern matches. Our methods allow us to give new proofs of several previously recorded results on this topic as well as to prove new extensions and new q-analogues of such results.     

Permutations and words counted by consecutive patterns

Generating functions which count occurrences of consecutive sequences in a permutation or a word which matches a given pattern are studied by exploiting the combinatorics associated with symmetric functions. Our theorems take the generating function for the number of permutations which do not contain a certain pattern and give generating functions refining permutations by the both the total number of pattern matches and the number of non-overlapping pattern matches. Our methods allow us to give new proofs of several previously recorded results on this topic as well as to prove new extensions and new q-analogues of such results.     

Multidimensional large deviation local limit theorems

An earlier paper by the author ([4], 97–114) established large deviation local limit theorems for arbitrary sequences of real valued random variables. This work showed clearly the connection between the Cramér series and large deviation rates. In this article we present large deviation local limit theorems for arbitrary multidimensional random variables based solely on conditions imposed on their moment generating functions. These results generalize the theorems of [12], 100–106) for sums of independent and identically distributed random vectors.     

指数有界的C余弦算子函数

本文引入了指数有界的Ｃ余弦算子函数的生成元，讨论了生成元的基本性质，建立了相应的生成定理、逼近定理及扰动定理．最后给出了指数有界的Ｃ余弦算子函数与指数有界的Ｃ半群的基本联系．     

On the unimodality of spherically symmetric stable distribution functions

Several theorems are obtained concerning the unimodality of spherically symmetric distribution functions. These theorems are used to show that a class of spherically symmetric infinitely divisible distribution functions that contains the class of spherically symmetric stable distribution functions is unimodal.     

Rational Approximations of Riemann--Liouville and Weyl Fractional Integrals

We obtain exact rational approximation orders for functions expressible as Riemann--Liouville and Weyl fractional integrals. New results and the strengthening and generalization of theorems due to Popov, Petrusheva, Pekarskii, Rusak, and the author, which are well known in the theory of rational approximation of differentiable functions, are obtained as consequences of theorems due to Pekarskii related to rational approximation of functions from the Hardy--Sobolev classes in the unit disk.     

Some extensions of Carlitz on cyclic sums and generating functions

Recently, the author (Proc. Amer. Math. Soc., 57 1976, 271–275) derived two theorems involving double series, which gave as a consequence new and known generating functions for the Jacobi polynomial. The method of proof differed from that of previous workers. Using an extension of this procedure, we present in this paper two theorems for double and m-dimensional series which generalize our previous work. These formulas also yield new generating functions for the Jacobi polynomial and extend some formulas of Carlitz (Boll. U.M.I. (3), 16 1961, 150–155) and others. A feature of this work is the inclusion of the Jacobi polynomial within the framework of m-dimensional cyclic sums, thus generalizing a main result of Carlitz (SIAM Rev., 6 1964, 20–30).     

Limit shapes of young diagrams. Two elementary approaches

Techniques are presented for obtaining the limit shapes of Young diagrams with respect to multiplicative measures, which arise in statistical mechanics. The approach employs neither complex analysis nor Tauberian theorems. Also, the limit shape is found for bounded and unbounded partitions with respect to the uniform measure, without using even generating functions. Bibliography: 6 titles.     

The rise and fall of the vector epsilon algorithm

The performance of the vector epsilon algorithm is governed by two important mathematical theorems which are briefly reviewed in context. We note that the performance of the vector epsilon algorithm is inevitably qualitatively incorrect for sequences whose generating functions have poles near unity. This difficulty is avoided by the use of hybrid vector Padé approximants.     

一类强极限定理和矩母函数方法

In virtue of the notion of likelihood ratio and moment generating function,the limit properties of the sequences of absolutely continuous random variables are studied,and a class of strong limit theorems represented by inequalities with random bounds are obtained.