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
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
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
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
By virtue of the normal ordering of vacuum projector we directly derive some new complicated operatoridentities, regarding to the generalized Stirling number.
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
In this paper, we derive the complete asymptotic expansion of classical Baskakov operators Vn (f;x) in the form of all coefficients of n^-k, k = 0, 1... being calculated explic- itly in terms of Stirling number of t...In this paper, we derive the complete asymptotic expansion of classical Baskakov operators Vn (f;x) in the form of all coefficients of n^-k, k = 0, 1... being calculated explic- itly in terms of Stirling number of the first and second kind and another number G(i, p). As a corollary, we also get the Voronovskaja-type result for the operators.展开更多
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 Schrödinger equation to Pöschl-Teller potentials.展开更多
In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix method...In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.展开更多
In the paper,with the help of the Fa′a di Bruno formula and an identity of the Bell polynomials of the second kind,the authors define degenerateλ-array type polynomials,establish two explicit formulas,and present se...In the paper,with the help of the Fa′a di Bruno formula and an identity of the Bell polynomials of the second kind,the authors define degenerateλ-array type polynomials,establish two explicit formulas,and present several recurrence relations of degenerateλ-array type polynomials and numbers.展开更多
In this paper,using the Jordan canonical form of the Pascal matrix Pn,we present a new approach for inverting the Pascal matrix plus a scalar Pn+aIn for arbitrary real number a≠1.
This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally...This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summarion formula is given展开更多
We prove some congruence relations satisfied by integral Stirling type pairs. The results settle a question posed by Hsu . In particular, they extend known congruence properties of Stirling numbers of the first kind ...We prove some congruence relations satisfied by integral Stirling type pairs. The results settle a question posed by Hsu . In particular, they extend known congruence properties of Stirling numbers of the first kind and the second kind.展开更多
The authors establish an explicit formula for the generalized Euler NumbersE2n^(x), and obtain some identities and congruences involving the higher'order Euler numbers, Stirling numbers, the central factorial numbe...The authors establish an explicit formula for the generalized Euler NumbersE2n^(x), and obtain some identities and congruences involving the higher'order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.展开更多
In this paper, using generating functions and Riordan arrays, we get some identities relating Genocchi numbers with Stirling numbers and Cauchy numbers.
This paper gives a unified approach to Hsu's two classes of extended GSN pairs in the setting of Hsu-Riordan partial monoid which is a generalization of Shapiro's Riordan group, and moreover Hsu-Wang transfer ...This paper gives a unified approach to Hsu's two classes of extended GSN pairs in the setting of Hsu-Riordan partial monoid which is a generalization of Shapiro's Riordan group, and moreover Hsu-Wang transfer theorem, Drown-Sprugnoli transfer formula and generalized Brown transfer lemma which display some transfer methods of different kinds of Hsu-Riordau arrays and identities respectively.展开更多
In this paper,we give several identities of finite sums and some infinite series involving powers and inverse of binomial coefficients,which extends the results of T.Trif.
EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn i...EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix T, is obtained, so an open urnblem urouosed bv EI-Mikkawv[2] is solved. Some combinatorial identities are also given.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.11175113)
文摘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.
文摘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).
基金supported in part by the National Natural Science Foundation of China(Grant No.12061033)by the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region(Grants No.NJZY20119)by the Natural Science Foundation of Inner Mongolia(Grant No.2019MS01007),China.
文摘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.
基金the Mathematical Tianyuan Foundation (Grant No.A0324645) of China
文摘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.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10874174 and 10947017/A05 the Specialized Research Fund for the Doctorial Progress of Higher Education of China under Grant No.20070358009
文摘By virtue of the normal ordering of vacuum projector we directly derive some new complicated operatoridentities, regarding to the generalized Stirling number.
基金funded by Research Deanship at the University of Ha’il,Saudi Arabia,through Project No.RG-21144.
文摘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.
基金Supported by the Natural Science Foundation of Beijing(1072006)
文摘In this paper, we derive the complete asymptotic expansion of classical Baskakov operators Vn (f;x) in the form of all coefficients of n^-k, k = 0, 1... being calculated explic- itly in terms of Stirling number of the first and second kind and another number G(i, p). As a corollary, we also get the Voronovskaja-type result for the operators.
文摘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 Schrödinger equation to Pöschl-Teller potentials.
文摘In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.
基金The first two authors,Mrs.Lan Wu and Xue-Yan Chen,were partially supported by the College Scientific Research Project of Inner Mongolia(Grant No.NJZY19156 and Grant No.NJZZ19144)by the Natural Science Foundation Project of Inner Mongolia(Grant No.2021LHMS05030)by the Development Plan for Young Technological Talents in Colleges and Universities of Inner Mongolia(Grant No.NJYT22051)in China.
文摘In the paper,with the help of the Fa′a di Bruno formula and an identity of the Bell polynomials of the second kind,the authors define degenerateλ-array type polynomials,establish two explicit formulas,and present several recurrence relations of degenerateλ-array type polynomials and numbers.
基金Supported by the Natural Science Foundation of Gansu Proveince(1010RJZA049)
文摘In this paper,using the Jordan canonical form of the Pascal matrix Pn,we present a new approach for inverting the Pascal matrix plus a scalar Pn+aIn for arbitrary real number a≠1.
文摘This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summarion formula is given
文摘We prove some congruence relations satisfied by integral Stirling type pairs. The results settle a question posed by Hsu . In particular, they extend known congruence properties of Stirling numbers of the first kind and the second kind.
基金the Guangdong Provincial Natural Science Foundation (No.05005928)the National Natural Science Foundation (No.10671155) of P.R.China
文摘The authors establish an explicit formula for the generalized Euler NumbersE2n^(x), and obtain some identities and congruences involving the higher'order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.
文摘In this paper, using generating functions and Riordan arrays, we get some identities relating Genocchi numbers with Stirling numbers and Cauchy numbers.
文摘This paper gives a unified approach to Hsu's two classes of extended GSN pairs in the setting of Hsu-Riordan partial monoid which is a generalization of Shapiro's Riordan group, and moreover Hsu-Wang transfer theorem, Drown-Sprugnoli transfer formula and generalized Brown transfer lemma which display some transfer methods of different kinds of Hsu-Riordau arrays and identities respectively.
基金Supported by the National Natural Science Foundation of China (Grant No. 11061020)the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant No. 20080404MS010)
文摘In this paper,we give several identities of finite sums and some infinite series involving powers and inverse of binomial coefficients,which extends the results of T.Trif.
基金the NSF of Gansu Province of China (3ZS041-A25-007)
文摘EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix T, is obtained, so an open urnblem urouosed bv EI-Mikkawv[2] is solved. Some combinatorial identities are also given.