In this paper, the author gives a simpler alternative approach to the LU factorization and 1-banded factorization of the Vandermonde matrix, and obtains explicit formulas of the triangular factors and 1-banded matrices by using symmetric functions.
In this paper, using the production matrix of an exponential Riordan array [g(t), f (t)], we give a recurrence relation for the Sheffer sequence for the ordered pair (g(t), f (t)). We also develop a new determinant representation for the general term of the Sheffer sequence. As applications, determinant expressions for some classical Sheffer polynomial sequences are derived.
Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turn 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.
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