若干组合序列的矩阵研究
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摘要
本文试图在经典组合序列与矩阵技术之间的联系上做些工作。具体内容如下:
1.研究了二项式系数(α-κ n-κ)、(α/(α+βn)(α+βn n)、(n+λ κ+λ)、满足条件α_(n,κ)=α_(n-1,κ-1)+α_(n-1,κ)的序列α_(n,κ)及它们所组成的矩阵及性质。得到了一些有价值的组合恒等式。
2.研究了Lab数ι(n,κ)=(n κ)(n-1)!/(κ-1)!及由它所组成的矩阵L_n,建立了Lah矩阵与Pascal矩阵、Stirling矩阵间的联系,得到了Lah矩阵的乘积分解与幂和形式。
3.不少学者研究了形如(i j)φ_(i-j)(x)所组成的矩阵及性质,其中φ_ n(x)有x~n,x~(n|λ)等形式。本文研究了更一般的矩阵L_n[x]=(ι(i,j)φ_(i-j)(x))_(n×n),其中ι(n,κ),φ_n(x)分别满足:
ι(i,κ)ι(κ,j)=ι(i,j)(i-j κ-j),φ_n(x+y)=sum from κ=0 to n(n κ)φ_κ(x)φ_(n-κ)(y)实际上,φ_n(x)就是二项式型多项式。得到的结果更具一般性。
4.Fibonacci序列更是由于其历史悠久,应用广泛而引入如胜。本文研究了二次线性递推序列(或称双变量Fibonacci序列):
F_(n+1)(α,λ,ι)=αF_n(α,λ,ι)+λF_(n-ι)(α,λ,ι)_n≥ι
考察了几种特殊情形,得到了一些相关组合恒等式。
构造了一个广义的Tribonacci矩阵:A=(x 1 0 y 0 1 z 0 0)。研究了三次线性递推序列:
T_(n+1)=xT_n+yT_(n-1)+zT_(n-2),T_0=0,T_1=1,T_2=x
通过二次线性递推序列的几种特殊情况和建立三次线性递推序列与Fibonacci数的联系,得到了一些组合恒等式及关于Fibonacci数的有趣表达式。
We intend to investigate the relations between the classical combinatorial sequences and their matrices in this paper. They are listed as follows:1. The binomial coefficients and sequence an,k such that, aan,k=an-1,k-1+an-1,k are studied. The corresponding matrices and their properties are investigated. Some valuable combinatorial identities are derived.2. Lah number and its matrix Ln is studied. The relations among Lah matrices, Pascal matrices and Stirling matrices are obtained. The factorizations and power sum form on Lah matrices are given.3. The matrices with elements and their properties have been studied carefully by many mathematicians, where ψn(x) are xn,xn|λ and other special forms respectively. In this paper, the matrices Ln[x] = (l(i,j)ψi-j(x))n×n with more general elements are investigated, where l(n,k) and ψn(x) satisfy the following respectively:In fact, ψn(x) is the polynomial of binomial type. Therefore, the results we obtained are more general.4. Fibonacci sequences are of long history and their applications appear in many fields. The following bivariate Fibonacci sequences (polynomials) are studied:Some special cases of two-order linear recurrences are investigated and related combinatorial identities are given.
A generalized Tribonacci matrix is constructed. The Tribonaccisequences {Tn} defined by: Tn+1= xTn+ yTn-1+zTn-2,T0=0,T1=1,T2=x are studied by the Tribonacci matrix. A relation between third-order linear recurrence and Fibonacci numbers is built, and some interesting expressions on Fibonacci numbers are derived.
1.研究了二项式系数(α-κ n-κ)、(α/(α+βn)(α+βn n)、(n+λ κ+λ)、满足条件α_(n,κ)=α_(n-1,κ-1)+α_(n-1,κ)的序列α_(n,κ)及它们所组成的矩阵及性质。得到了一些有价值的组合恒等式。
2.研究了Lab数ι(n,κ)=(n κ)(n-1)!/(κ-1)!及由它所组成的矩阵L_n,建立了Lah矩阵与Pascal矩阵、Stirling矩阵间的联系,得到了Lah矩阵的乘积分解与幂和形式。
3.不少学者研究了形如(i j)φ_(i-j)(x)所组成的矩阵及性质,其中φ_ n(x)有x~n,x~(n|λ)等形式。本文研究了更一般的矩阵L_n[x]=(ι(i,j)φ_(i-j)(x))_(n×n),其中ι(n,κ),φ_n(x)分别满足:
ι(i,κ)ι(κ,j)=ι(i,j)(i-j κ-j),φ_n(x+y)=sum from κ=0 to n(n κ)φ_κ(x)φ_(n-κ)(y)实际上,φ_n(x)就是二项式型多项式。得到的结果更具一般性。
4.Fibonacci序列更是由于其历史悠久,应用广泛而引入如胜。本文研究了二次线性递推序列(或称双变量Fibonacci序列):
F_(n+1)(α,λ,ι)=αF_n(α,λ,ι)+λF_(n-ι)(α,λ,ι)_n≥ι
考察了几种特殊情形,得到了一些相关组合恒等式。
构造了一个广义的Tribonacci矩阵:A=(x 1 0 y 0 1 z 0 0)。研究了三次线性递推序列:
T_(n+1)=xT_n+yT_(n-1)+zT_(n-2),T_0=0,T_1=1,T_2=x
通过二次线性递推序列的几种特殊情况和建立三次线性递推序列与Fibonacci数的联系,得到了一些组合恒等式及关于Fibonacci数的有趣表达式。
We intend to investigate the relations between the classical combinatorial sequences and their matrices in this paper. They are listed as follows:1. The binomial coefficients and sequence an,k such that, aan,k=an-1,k-1+an-1,k are studied. The corresponding matrices and their properties are investigated. Some valuable combinatorial identities are derived.2. Lah number and its matrix Ln is studied. The relations among Lah matrices, Pascal matrices and Stirling matrices are obtained. The factorizations and power sum form on Lah matrices are given.3. The matrices with elements and their properties have been studied carefully by many mathematicians, where ψn(x) are xn,xn|λ and other special forms respectively. In this paper, the matrices Ln[x] = (l(i,j)ψi-j(x))n×n with more general elements are investigated, where l(n,k) and ψn(x) satisfy the following respectively:In fact, ψn(x) is the polynomial of binomial type. Therefore, the results we obtained are more general.4. Fibonacci sequences are of long history and their applications appear in many fields. The following bivariate Fibonacci sequences (polynomials) are studied:Some special cases of two-order linear recurrences are investigated and related combinatorial identities are given.
A generalized Tribonacci matrix is constructed. The Tribonaccisequences {Tn} defined by: Tn+1= xTn+ yTn-1+zTn-2,T0=0,T1=1,T2=x are studied by the Tribonacci matrix. A relation between third-order linear recurrence and Fibonacci numbers is built, and some interesting expressions on Fibonacci numbers are derived.
引文
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