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A Family of Generalized Stirling Numbers of the First Kind
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作者 Beih S. El-Desouky Nabela A. El-Bedwehy +1 位作者 Abdelfattah Mustafa Fatma M. Abdel Menem 《Applied Mathematics》 2014年第10期1573-1585,共13页
A modified approach via differential operator is given to derive a new family of generalized Stirling numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky?[1]?and Gould?... A modified approach via differential operator is given to derive a new family of generalized Stirling numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky?[1]?and Gould?[2]. Some new combinatorial identities and many relations between different types of Stirling numbers are found. Furthermore, some interesting special cases of the generalized Stirling numbers of the first kind are deduced. Also, a connection between these numbers and the generalized harmonic numbers is derived. Finally, some applications in coherent states and matrix representation of some results obtained are given. 展开更多
关键词 stirling numbers Comtet numbers CREATION ANNIHILATION Differential Operator MAPLE Program
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An Equation of Stirling Numbers of the Second Kind 被引量:3
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作者 DU Chun-yu ZHANG Jian-guo 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2006年第2期261-263,共3页
In this paper, We propose and prove the following equation, where {3^n} is a stirling number of the second kind, when n ≥ 3 is given.
关键词 COMBINATIONS stirling numbers of the second kind
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Quantum mechanical operator realization of the Stirling numbers theory studied by virtue of the operator Hermite polynomials method
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作者 范洪义 楼森岳 《Chinese Physics B》 SCIE EI CAS CSCD 2015年第7期102-105,共4页
Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with s... Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials. 展开更多
关键词 operator Hermite polynomials method(OHPM) stirling numbers
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Some Properties of Degenerate r-Dowling Polynomials and Numbers of the Second Kind
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作者 Hye Kyung Kim Dae Sik Lee 《Computer Modeling in Engineering & Sciences》 SCIE EI 2022年第12期825-842,共18页
The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and o... The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics.In recent years,some mathematicians have studied degenerate version of them and obtained many interesting results.With this in mind,in this paper,we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind.We derive many interesting properties and identities for them including generating functions,Dobinski-like formula,integral representations,recurrence relations,differential equation and various explicit expressions.In addition,we explore some expressions for them that can be derived from repeated applications of certain operators to the exponential functions,the derivatives of them and some identities involving them. 展开更多
关键词 Dowling lattice Whitney numbers and polynomials r-Whitney numbers and polynomials of the second kind r-Bell polynomials r-stirling numbers dowling numbers and polynomials
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On Polynomials Rn(x) Related to the Stirling Numbers and the Bell Polynomials Associated with the p-Adic Integral on
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作者 Hui Young Lee Cheon Seoung Ryoo 《Open Journal of Discrete Mathematics》 2016年第2期89-98,共10页
In this paper, one introduces the polynomials R<sub>n</sub>(x) and numbers R<sub>n</sub> and derives some interesting identities related to the numbers and polynomials: R<sub>n</sub>... In this paper, one introduces the polynomials R<sub>n</sub>(x) and numbers R<sub>n</sub> and derives some interesting identities related to the numbers and polynomials: R<sub>n</sub> and R<sub>n</sub>(x). We also give relation between the Stirling numbers, the Bell numbers, the R<sub>n</sub> and R<sub>n</sub>(x). 展开更多
关键词 the Euler numbers and Polynomials the stirling numbers the Bell Polynomials and numbers
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用无穷矩阵方程求解第二类Stirling数表示的自然数幂和 被引量:1
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作者 唐军强 艾英 《高师理科学刊》 2023年第1期20-23,共4页
讨论了4个用第二类Stirling数表示的自然数的幂和公式.利用升阶乘和降阶乘的定义式,得到关于各阶幂和的递推关系,用求解无穷矩阵方程的方法给出用第二类Stirling数表示的幂和公式,并证明了它们之间的等价性.
关键词 无穷矩阵方程 自然数 幂和 第一类stirling 第二类stirling
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Product of Uniform Distribution and Stirling Numbers of the First Kind 被引量:6
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作者 Ping SUN 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2005年第6期1435-1442,共8页
Let Vk=u1u2……uk, ui's be i.i.d - U(0, 1), the p.d.f of 1 - Vk+l be the GF of the unsigned Stirling numbers of the first kind s(n, k). This paper discusses the applications of uniform distribution to combinator... Let Vk=u1u2……uk, ui's be i.i.d - U(0, 1), the p.d.f of 1 - Vk+l be the GF of the unsigned Stirling numbers of the first kind s(n, k). This paper discusses the applications of uniform distribution to combinatorial analysis and Riemann zeta function; several identities of Stifling series are established, and the Euler's result for ∑ Hn/n^k-l, k ≥ 3 is given a new probabilistic proof. 展开更多
关键词 stirling numbers generating function uniform distribution MOMENT Riemann zeta function
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ASYMPTOTIC APPROXIMATION OF FUNCTIONS AND THEIR DERIVATIVES BY GENERALIZED BASKAKOV-SZAZS-DURRMEYER OPERATORS 被引量:1
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作者 Ulrich Abel Vijay Gupta Mircea Ivan 《Analysis in Theory and Applications》 2005年第1期15-26,共12页
We present the complete asymptotic expansion for a generalization of the Baskakov-Szasz-Durrmeyer operators and their derivatives.
关键词 approximation by positive operators rate of convergence degree of approximation asymptotic expansion stirling numbers
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Analysis of the Multi-Pivot Quicksort Process 被引量:1
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作者 Mahmoud Ragab Beih El-Sayed El-Desouky Nora Nader 《Open Journal of Modelling and Simulation》 2017年第1期47-58,共12页
In this paper, we study a new version from Dual-pivot Quicksort algorithm when we have some other number of pivots. Hence, we discuss the idea of picking pivots ?by random way and splitting the list simultaneously acc... In this paper, we study a new version from Dual-pivot Quicksort algorithm when we have some other number of pivots. Hence, we discuss the idea of picking pivots ?by random way and splitting the list simultaneously according to these. The modified version generalizes these results for multi process. We show that the average number of swaps done by Multi-pivot Quicksort process and we present a special case. Moreover, we obtain a relationship between the average number of swaps of Multi-pivot Quicksort and Stirling numbers of the first kind. 展开更多
关键词 QUICKSORT Convergence Multi-Pivot QUICKSORT PROCESS stirling Number of the first kind
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Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities
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作者 Siqintuya Jin Bai-Ni Guo Feng Qi 《Computer Modeling in Engineering & Sciences》 SCIE EI 2022年第9期781-799,共19页
In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extende... In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory. 展开更多
关键词 Connection EQUIVALENCE closed-form formula combinatorial identity partial Bell polynomial falling factorial rising factorial binomial coefficient stirling number of the first kind stirling number of the second kind problem
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The Generalized <i>r</i>-Whitney Numbers
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作者 B. S. El-Desouky F. A. Shiha Ethar M. Shokr 《Applied Mathematics》 2017年第1期117-132,共16页
In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating fun... In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of the relations between Whitney and Stirling numbers is given. 展开更多
关键词 Whitney numbers r-Whitney numbers p-stirling numbers GENERALIZED q-stirling numbers GENERALIZED stirling numbers
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New Extension of Unified Family Apostol-Type of Polynomials and Numbers
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作者 Beih El-Sayed El-Desouky Rabab Sabry Gomaa 《Applied Mathematics》 2015年第9期1495-1505,共11页
The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in [1] [2]. Also, we derive some properties for these polynomi... The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type polynomials and numbers based on results given in [1] [2]. Also, we derive some properties for these polynomials and obtain some relationships between the Jacobi polynomials, Laguerre polynomials, Hermite polynomials, Stirling numbers and some other types of generalized polynomials. 展开更多
关键词 Euler BERNOULLI and Genocchi POLYNOMIALS stirling numbers LAGUERRE POLYNOMIALS Hermite POLYNOMIALS
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Generalized Legendre-Stirling Numbers
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作者 K. C. Garrett Kendra Killpatrick 《Open Journal of Discrete Mathematics》 2014年第4期109-114,共6页
The Legendre-Stirling numbers were discovered by Everitt, Littlejohn and Wellman in 2002 in a study of the spectral theory of powers of the classical second-order Legendre differential operator. In 2008, Andrews and L... The Legendre-Stirling numbers were discovered by Everitt, Littlejohn and Wellman in 2002 in a study of the spectral theory of powers of the classical second-order Legendre differential operator. In 2008, Andrews and Littlejohn gave a combinatorial interpretation of these numbers in terms of set partitions. In 2012, Mongelli noticed that both the Jacobi-Stirling and the Legendre-Stirling numbers are in fact specializations of certain elementary and complete symmetric functions and used this observation to give a combinatorial interpretation for the generalized Legendre-Stirling numbers. In this paper we provide a second combinatorial interpretation for the generalized Legendre-Stirling numbers which more directly generalizes the definition of Andrews and Littlejohn and give a combinatorial bijection between our interpretation and the Mongelli interpretation. We then utilize our interpretation to prove a number of new identities for the generalized Legendre-Stirling numbers. 展开更多
关键词 stirling numbers Legendre-stirling numbers SET PARTITIONS
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Operator Formulas Involving Generalized Stirling Number Derived by Virtue of Normal Ordering of Vacuum Projector
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作者 范洪义 姜年权 《Communications in Theoretical Physics》 SCIE CAS CSCD 2010年第10期651-653,共3页
By virtue of the normal ordering of vacuum projector we directly derive some new complicated operatoridentities, regarding to the generalized Stirling number.
关键词 operator formulas generalized stirling number normal operator
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Some New Results on the Number of Paths
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作者 Beih S. El-Desouky Abdelfattah Mustafa E. M. Mahmoud 《Open Journal of Modelling and Simulation》 2015年第3期63-69,共7页
Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array n*m denoted by ?Anm. We investigate the g... Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array n*m denoted by ?Anm. We investigate the generating function for the general case and hence some special cases as well. The probability function of the number of paths along is obtained. Moreover, the moment generating function of the random variable X and hence the mean and variance are obtained. Finally, some applications are introduced. 展开更多
关键词 stirling numbers GENERATING FUNCTION Moment GENERATING FUNCTION Comtet numbers MAPLE Program
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Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials
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作者 Alfred Wünsche 《Advances in Pure Mathematics》 2017年第2期213-261,共49页
Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting proper... Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schr&ouml;dinger equation to P&ouml;schl-Teller potentials. 展开更多
关键词 Orthogonal Polynomials Lie Algebra SU(1 1) and Lie Group SU(1 1) Lowering and Raising Operators Jacobi Polynomials Ultraspherical Polynomials Gegenbauer Polynomials Chebyshev Polynomials Legendre Polynomials stirling numbers Hypergeometric Function Operator Identities Vandermond’s Convolution Identity Poschl-Teller Potentials
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A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable
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作者 N.Alam W.A.Khan +5 位作者 S.Obeidat G.Muhiuddin N.S.Diab H.N.Zaidi A.Altaleb L.Bachioua 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第4期187-209,共23页
In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions... In this article,we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials.Some fundamental properties of these functions are given.By using these generating functions and some identities,relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials,Stirling numbers are presented.Computational formulae for these polynomials are obtained.Applying a partial derivative operator to these generating functions,some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained.In addition,some remarks and observations on these polynomials are given. 展开更多
关键词 Bernoulli polynomials euler polynomials bell polynomials stirling numbers
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联系Bernoulli数和第二类Stirling数的一个恒等式 被引量:7
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作者 褚维盘 党四善 《纯粹数学与应用数学》 CSCD 2004年第3期282-284,共3页
利用指数型生成函数建立起联系 Bernoulli数和第二类
关键词 指数型生成函数 BERNOULLI数 第二类stirling 恒等式
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一些包含Chebyshev多项式和Stirling数的恒等式 被引量:4
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作者 刘国栋 罗辉 《纯粹数学与应用数学》 CSCD 2010年第2期177-182,共6页
利用初等方法研究Chebyshev多项式的性质,建立了广义第二类Chebyshev多项式的一个显明公式,并得到了一些包含第一类Chebyshev多项式,第一类Stirling数和Lucas数的恒等式.
关键词 第一类CHEBYSHEV多项式 第二类CHEBYSHEV多项式 第一类stirling FIBONACCI数 LUCAS数
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涉及Euler数、Bernoulli数和推广的第一类Stirling数的一些恒等式 被引量:5
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作者 党四善 褚维盘 《纯粹数学与应用数学》 CSCD 1997年第2期109-113,117,共6页
利用递推关系把文[1]、[2]中的有关结论推广到一般情形,建立起涉及Eu-ler数、Bernouli数和推广的第一类Stirling数的一些恒等式.
关键词 stirling 恒等式 递推关系 欧拉数 伯努利数
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