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Sheng-liang Yang

^{}《Discrete Applied Mathematics》2008,156(15):3040-3045In this paper, we study the Jordan canonical form of the generalized Pascal functional matrix associated with a sequence of binomial type, and demonstrate that the transition matrix between the generalized Pascal functional matrix and its Jordan canonical form is the iteration matrix associated with the binomial sequence. In addition, some combinatorial identities are derived from the corresponding matrix factorization. 相似文献

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Using the generating functions, we prove some symmetry identities for the Euler polynomials and higher order Euler polynomials, which generalize the multiplication theorem for the Euler polynomials. Also we obtain some relations between the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers. 相似文献

3.

Sheng-liang Yang Sai-nan Zheng Shao-peng Yuan Tian-Xiao He 《Linear algebra and its applications》2013

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers. 相似文献

4.

Sheng-liang Yang 《Discrete Mathematics》2008,308(1):51-58

Using the exponential generating function and the Bell polynomials, we obtain several new identities for the binomial sequences. As applications, some interesting identities are established for the Abel polynomials, exponential polynomials and factorial powers. 相似文献

5.

Sheng-liang Yang 《Discrete Mathematics》2008,308(4):550-554

The main purpose of this paper is to prove an identity of symmetry for the higher order Bernoulli polynomials. It turns out that the recurrence relation and multiplication theorem for the Bernoulli polynomials which discussed in [F.T. Howard, Application of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995) 157-172], as well as a relation of symmetry between the power sum polynomials and the Bernoulli numbers developed in [H.J.H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer. Math. Monthly 108 (2001) 258-261], are all special cases of our results. 相似文献

6.

Denote 〈

*x*|*d*〉_{ n }=*x*(*x*+*d*)(*x*+ 2*d*) · · · (*x*+ (*n*- 1)*d*) for*n*= 1, 2, · · ·, and 〈*x*|*d*〉_{0}= 1, where 〈*x*|*d*〉_{ n }is called the generalized factorial of*x*with increment*d*. In this paper, we present the evaluation of Hankel determinants of sequence of generalized factorials. The main tool used for the evaluation is the method based on exponential Riordan arrays. Furthermore, we provide Hankel determinant evaluations of the Eulerian polynomials and exponential polynomials. 相似文献7.

For integers It is known that the (

相似文献*m*>*r*≥ 0, Brietzke (2008) defined the (*m*,*r*)-central coefficients of an infinite lower triangular matrix*G*= (*d*,*h*) = (*d*_{n,k})_{n,k∈N}as*d*_{mn+r,(m−1)n+r}, with*n*= 0, 1, 2,..., and the (*m*,*r*)-central coefficient triangle of*G*as$${G^{\left( {m,r} \right)}} = {\left( {{d_{mn + r,\left( {m - 1} \right)n + k + r}}} \right)_{n,k \in \mathbb{N}}}.$$

*m*,*r*)-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array*G*= (*d*,*h*) with*h*(0) = 0 and*d*(0),*h*′(0) ≠ 0, we obtain the generating function of its (*m*,*r*)-central coefficients and give an explicit representation for the (*m*,*r*)-central Riordan array*G*^{(m,r)}in terms of the Riordan array*G*. Meanwhile, the algebraic structures of the (*m*,*r*)-central Riordan arrays are also investigated, such as their decompositions, their inverses, and their recessive expressions in terms of*m*and*r*. As applications, we determine the (*m*,*r*)-central Riordan arrays of the Pascal matrix and other Riordan arrays, from which numerous identities are constructed by a uniform approach.8.

Sheng-Liang Yang 《Discrete Applied Mathematics》2007,155(15):2025-2030

In this paper, we are going to study some additional relations between the Stirling matrix

*S*_{n}and the Pascal matrix*P*_{n}. Also the representation for the matrix*T*_{n}and in terms of*s*_{n}and*S*_{n}will be considered. Consequently, this will give an answer to an open problem proposed by EI-Mikkawy [On a connection between the Pascal, Vandermonde and Stirling matrices—II, Appl. Math. Comput. 146 (2003) 759-769]. 相似文献9.

Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form ${(z+b)}^{n}$. 相似文献

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In this paper, we define a class of strongly connected digraph, called the k-walk- regular digraph, study some properties of it, provide its some algebraic characterization and point out that the 0-walk-regular digraph is the same as the walk-regular digraph discussed by Liu and Lin in 2010 and the D-walk-regular digraph is identical with the weakly distance-regular digraph defined by Comellas et al in 2004. 相似文献

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