AM  Vol.6 No.6 , June 2015
An Algorithm to Generalize the Pascal and Fibonacci Matrices
Author(s) Ilhan M. Izmirli
ABSTRACT
The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.

Cite this paper
Izmirli, I. (2015) An Algorithm to Generalize the Pascal and Fibonacci Matrices. Applied Mathematics, 6, 1107-1114. doi: 10.4236/am.2015.66101.
References
[1]   Edwards, A.W.F. (2002) Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea. John Hopkins University Press, Baltimore.

[2]   Edwards, A.W.F. (2013) The Arithmetical Triangle. In: Wilson, R. and Watkins, J.J., Eds., Combinatorics: Ancient and Modern, Oxford University Press, Oxford, 166-180.
http://dx.doi.org/10.1093/acprof:oso/9780199656592.003.0008

[3]   Coolidge, J.L. (1949) The Story of the Binomial Theorem. The American Mathematical Monthly, 56, 147-157.
http://dx.doi.org/10.2307/2305028

[4]   Smith, K.J. (2010) Nature of Mathematics, Cengage Learning.

[5]   Call, G.S. and Velleman, D.J. (1993) Pascal’s Matrices. American Mathematical Monthly, 100, 372-376.
http://dx.doi.org/10.2307/2324960

[6]   Alan, E. and Strang, G. (2004) Pascal Matrices. American Mathematical Monthly, 111, 361-385.

[7]   Goonatilake, S. (1998) Toward a Global Sciece. Indiana University Press, Bloomington.

[8]   Siegler, L.E. (2002) Fibonacci’s Liber Abaci: A Translation into Modern English of the Book of Calculation, Sources and Studies in the History of Mathematics and Physical Sciences. Springer, Berlin.
http://dx.doi.org/10.1007/978-1-4613-0079-3

[9]   Hoggartt Jr., V.E. and Bricknell, M. (1974) Triangular Numbers. The Fibonacci Quarterly, 12, 221-230.

 
 
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