ABSTRACT
Presently, a growing interest is observed in development of methods using a combinatorial analysis based on conceptions and objects from modern combinatorics, Vajda, 1989, Bergum, 1994, Ross, 1996. In a field of engineering problems various structures of the so-called numerical triangles and hyperbolic Fibonacci functions can be used for modeling and numerical analysis of distributed parameter systems, Trzaska, 1997, Rydygier and Trzaska, 1999. Monic non-zero polynomials which generate the first modified numerical triangle, FMNT, are defined by the following recurrence, Trzaska, 1993a, 1993b
Tn+2(x) = (2 + x)Tn+1(x)-Tn(x), n = 0, 1, 2, ... , (2.1) with TQ(X) — 1 and T\(x) = 1 + x as initial elements. From the above recurrence, the following polynomials can be calculated
Thus, the polynomial Tn(x) can be written in the form n
Tn(x) = Y, 0",***. « = 0, 1, 2, ... (2.2) k=0
where the coefficients on j&,n = 0, 1, 2, ..., 0 < k < n, fulfill the relation
an,k — 2an_i,A, + an-i,k-i - an-2,k (2.3)
with 00,0 = 1 and 0,1$ = 1 as initial values. Based on (3) the FMNT can be constructed. It is presented in Table 1.
