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Informace o vydavateli: Mathematical Association of America (MAA), Washington, D.C.; Taylor \& Francis, Abingdon, Oxfordshire
Témata: Matrix exponential and similar functions of matrices, Combinatorial probability, Factorials, binomial coefficients, combinatorial functions, Matrices of integers
Popis: The \(n \times n\) Pascal matrix \(P=(p_{ij})\) is obtained by taking the first \(n\) rows of Pascal's triangle and filling in with 0's on the right, i.e. \[ p_{ij}= \begin{cases} {i-1 \choose j-1}, \quad & \text{if } i \geq j \\ 0, & \text{otherwise}.\end{cases} \] For any nonzero real number \(x\), let the matrix \(P[x]=(p_{ij}(x))\) be defined as follows \[ p_{ij} (x)= \begin{cases} x^{i-j}{i-1 \choose j-1}, \quad & \text{if } i \geq j \\ 0, & \text{otherwise}\end{cases} \] and let \(P[0]\) be the identity matrix. For any square matrix \(A\), the exponential of \(A\) is defined to be the matrix \[ e^ A=I+A+{A^ 2 \over 2!}+{A^ 3 \over 3!}+\cdots+{A^ k \over k!}+ \cdots. \] The authors show that for every real number \(x\), \(P[x]=e^{xL}\), where \(L=(\ell_{ij})\) is defined by \[ \ell_{ij}=\begin{cases} j, \quad & \text{if } i=j+1 \\ 0, & \text{otherwise}.\end{cases} \]
Druh dokumentu: Article
Popis souboru: application/xml
DOI: 10.2307/2324960
Přístupová URL adresa: https://zbmath.org/203292
Přístupové číslo: edsair.c2b0b933574d..a6539dbbcf71b9ba85f3e970cabe8572
Databáze: OpenAIRE
Popis
Abstrakt:The \(n \times n\) Pascal matrix \(P=(p_{ij})\) is obtained by taking the first \(n\) rows of Pascal's triangle and filling in with 0's on the right, i.e. \[ p_{ij}= \begin{cases} {i-1 \choose j-1}, \quad & \text{if } i \geq j \\ 0, & \text{otherwise}.\end{cases} \] For any nonzero real number \(x\), let the matrix \(P[x]=(p_{ij}(x))\) be defined as follows \[ p_{ij} (x)= \begin{cases} x^{i-j}{i-1 \choose j-1}, \quad & \text{if } i \geq j \\ 0, & \text{otherwise}\end{cases} \] and let \(P[0]\) be the identity matrix. For any square matrix \(A\), the exponential of \(A\) is defined to be the matrix \[ e^ A=I+A+{A^ 2 \over 2!}+{A^ 3 \over 3!}+\cdots+{A^ k \over k!}+ \cdots. \] The authors show that for every real number \(x\), \(P[x]=e^{xL}\), where \(L=(\ell_{ij})\) is defined by \[ \ell_{ij}=\begin{cases} j, \quad & \text{if } i=j+1 \\ 0, & \text{otherwise}.\end{cases} \]
DOI:10.2307/2324960