In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as...In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived.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
An independent method for paper [10] is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths. Convolu...An independent method for paper [10] is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths. Convolutions for path counts are investigated, which yields some Vandcrmondc-type identities for multinomial and q-multinomial coefficients.展开更多
In this paper,we deduce several new identities on infinite series with denominators of summands containing both binomial coefficients and linear parts.For example,we evaluate the sums^(∞)∑_(k=1)x^(k)_(0)/(2k-1)(^(3k...In this paper,we deduce several new identities on infinite series with denominators of summands containing both binomial coefficients and linear parts.For example,we evaluate the sums^(∞)∑_(k=1)x^(k)_(0)/(2k-1)(^(3k)_(k))and^(∞)∑_(k=0)x^(k)_(0)/(3k+2)(^(3k)_(k))for any x_(0)∈(-27/4,27/4).For any 1<n≤85/4,we obtain the following fast converging series for log n:(∞)∑_(k=0)(2(n^(2)+6n+1)^(2)k+n^(4)+30n^(2)+1)(n-1)^(4k)/(4k+1)(-n)^(k)(n+1)^(2k)(^(4k)_(2k))=8n(n+1)^(2)-2n(n^(2)-1)log n.In addition,we pose many new conjectural series identities involving binomial coefficients;for example,we conjecture that(∞)∑_(k=1)9(21k-8)H^((4))_(k-1)+25/k^(3)/k^(3)(^(2k)_(k))^(3)=13π^(6)/3780,where H^((4))_(k-1)denotes the fourth harmonic number∑_(0<≤k-1)j^(-4).展开更多
Here introduced and studied are two formulaic classes consisting of various combinatorial algebraic identities and series summation formulas. The basic ideas include utilizing properly the △-operator and Stirling num...Here introduced and studied are two formulaic classes consisting of various combinatorial algebraic identities and series summation formulas. The basic ideas include utilizing properly the △-operator and Stirling numbers for some series transformations. A variety of classic formulas and remarkable identities are shown to be the members of the classes.展开更多
Here investigated is a kind of nonlinear combinatorial expression involving Fibonacci numbers defined on the set of integers. A number of particular consequences will be presented as examples.
By applying the theory of formal power series,the author obtains the closed forms for two kinds of infinite series involving the reciprocals of binomial coefficients,and the author gets another closed form for the inf...By applying the theory of formal power series,the author obtains the closed forms for two kinds of infinite series involving the reciprocals of binomial coefficients,and the author gets another closed form for the infinite series Σr≥m tn＋r/（n＋rr）.展开更多
In this paper, by means of Gould-Hsu inverse series relations, we establish several Gould-Hsu inversion chains. As consequence, some new transformation formulae as well as some famous hypergeometric series identities ...In this paper, by means of Gould-Hsu inverse series relations, we establish several Gould-Hsu inversion chains. As consequence, some new transformation formulae as well as some famous hypergeometric series identities are derived.展开更多
In this paper, Jabotinsky matrices in [4, 5] are modified and a type of infinite lower triangular matrices T(f) is discussed. Some algebraic properties of T(f) are obtained and proved. Additionally, some inverse pairs...In this paper, Jabotinsky matrices in [4, 5] are modified and a type of infinite lower triangular matrices T(f) is discussed. Some algebraic properties of T(f) are obtained and proved. Additionally, some inverse pairs and combinatorial identities associated with derivatives are obtained.展开更多
By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E7 into a sum of irreducible s...By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E7 into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E7. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan＇s well-known fourth-order Ez-invariant partial differential equation.展开更多
C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, H...C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.展开更多
基金Supported by Zhoukou Normal University High-Level Talents Start-Up Funds Research Project(Grant No.ZKNUC2022007)the Postgraduate Research&Practice Innovation Program of Jiangsu Province(Grant No.KYCX240725).
文摘In this paper,we firstly establish a combinatorial identity with a free parameter x,and then by means of derivative operation,several summation formulae concerning classical and generalized harmonic numbers,as well as binomial coefficients are derived.
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.
文摘An independent method for paper [10] is presented. Weighted lattice paths are enumerated by counting function which is a natural extension of Gaussian multinomial coefficient in the case of unrestricted paths. Convolutions for path counts are investigated, which yields some Vandcrmondc-type identities for multinomial and q-multinomial coefficients.
基金Supported by the Natural Science Foundation of China(grant no.12371004)。
文摘In this paper,we deduce several new identities on infinite series with denominators of summands containing both binomial coefficients and linear parts.For example,we evaluate the sums^(∞)∑_(k=1)x^(k)_(0)/(2k-1)(^(3k)_(k))and^(∞)∑_(k=0)x^(k)_(0)/(3k+2)(^(3k)_(k))for any x_(0)∈(-27/4,27/4).For any 1<n≤85/4,we obtain the following fast converging series for log n:(∞)∑_(k=0)(2(n^(2)+6n+1)^(2)k+n^(4)+30n^(2)+1)(n-1)^(4k)/(4k+1)(-n)^(k)(n+1)^(2k)(^(4k)_(2k))=8n(n+1)^(2)-2n(n^(2)-1)log n.In addition,we pose many new conjectural series identities involving binomial coefficients;for example,we conjecture that(∞)∑_(k=1)9(21k-8)H^((4))_(k-1)+25/k^(3)/k^(3)(^(2k)_(k))^(3)=13π^(6)/3780,where H^((4))_(k-1)denotes the fourth harmonic number∑_(0<≤k-1)j^(-4).
文摘Here introduced and studied are two formulaic classes consisting of various combinatorial algebraic identities and series summation formulas. The basic ideas include utilizing properly the △-operator and Stirling numbers for some series transformations. A variety of classic formulas and remarkable identities are shown to be the members of the classes.
文摘Here investigated is a kind of nonlinear combinatorial expression involving Fibonacci numbers defined on the set of integers. A number of particular consequences will be presented as examples.
基金Supported by the NNSF of China(10771093)Supported by the NSF of Henan Province(0511010300)
文摘By applying the theory of formal power series,the author obtains the closed forms for two kinds of infinite series involving the reciprocals of binomial coefficients,and the author gets another closed form for the infinite series Σr≥m tn＋r/（n＋rr）.
基金Supported by the Science Foundation of Dalian Nationalities University (Grant No. DC110305)
文摘In this paper, by means of Gould-Hsu inverse series relations, we establish several Gould-Hsu inversion chains. As consequence, some new transformation formulae as well as some famous hypergeometric series identities are derived.
文摘In this paper, Jabotinsky matrices in [4, 5] are modified and a type of infinite lower triangular matrices T(f) is discussed. Some algebraic properties of T(f) are obtained and proved. Additionally, some inverse pairs and combinatorial identities associated with derivatives are obtained.
基金Supported by NSFC(Grant Nos.11171324 and 11321101)
文摘By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra E7 into a sum of irreducible submodules. This essentially gives a partial differential equation proof of a combinatorial identity on the dimensions of certain irreducible modules of E7. We also determine two three-parameter families of irreducible submodules in the solution space of Cartan＇s well-known fourth-order Ez-invariant partial differential equation.
基金the National Natural Science Foundation of China(Grant No.10471016)the Natural Science Foundation of Henan Province(Grant No.0511010300)the Natural Science Foundation of the Education Department of Henan Province(Grant No.200510482001)
文摘C. Radoux （J. Comput. Appl. Math., 115 （2000） 471-477） obtained a computational formula of Hankel determinants on some classical combinatorial sequences such as Catalan numbers and polynomials, Bell polynomials, Hermite polynomials, Derangement polynomials etc. From a pair of matrices this paper introduces two kinds of numbers. Using the first kind of numbers we give a unified treatment of Hankel determinants on those sequences, i.e., to consider a general representation of Hankel matrices on the first kind of numbers. It is interesting that the Hankel determinant of the first kind of numbers has a close relation that of the second kind of numbers.
