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The linear algebra of the generalized Pascal matrix. (English) Zbl 0873.15014

This paper is concerned with the “generalized Pascal matrices” \(P_n[x]\), \(Q_n[x]\), \(R_n[x]\) whose \((i,j)\)-entries are \(x^{i-j} {i\choose j}\), \(x^{i+j} {i\choose j}\), \(x^{i+j} {{i+j}\choose j}\) respectively \((i,j=0,1,\dots,n\); \(x\) is a variable). Generalizing results of R. Brawer and M. Pinovino [Linear Algebra Appl. 174, 13–23 (1992; Zbl 0755.15012)], the author expresses \(P_n[n]\) and \(Q_n[x]\) as products of lower unitriangular “summation matrices” and proves the Cholesky factorizations \(R_n[x]= Q_n[x] P_n^T[x]= P_n[x] Q_n^T[x]\).

MSC:

15B36 Matrices of integers
15A23 Factorization of matrices
65F05 Direct numerical methods for linear systems and matrix inversion

Citations:

Zbl 0755.15012
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References:

[1] Call, G. S.; Velleman, D. J., Pascal’s matrices, Amer. Math. Monthly, 100, 372-376 (1993) · Zbl 0788.05011
[2] Brawer, R.; Pirovino, M., The linear algebra of the Pascal matrix, Linear Algebra Appl., 174, 13-23 (1992) · Zbl 0755.15012
[3] Riordan, J., Combinatorial Identities (1968), Wiley: Wiley New York · Zbl 0194.00502
[4] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1980), Springer-Verlag: Springer-Verlag New York · Zbl 0423.65002
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