Explicit Formulæ for some Numerical Triangles of Z. W. Trzaska

Helmut Prodinger Department for Algebra and Discrete Mathematics TU Wien Wiedner Hauptstrasse 8–10/118 A-1040 Wien Austria email: Helmut.Prodinger@tuwien.ac.at

In [3], the following modified numerical triangle Mnt1 is considered. The polynomials T_{k}(q) are given by T_{0}(q) = 1, T_{1}(q) = 1 + q and the recursion formula

T_{k+1}(q) = (2 + q)T_{k}(q) − T_{k−1}(q) .  (1)

The numbers A(k, m) of the modified numerical triangle Mnt1 are defined as

A(k, m) = [q^{m}]T_{k}(q) .

(Read: “coefficient of q^{m} in T_{k}(q).”) Theorem 1.

A(k, m) = \binom{k + m}{2m} .

The second modified numerical triangle Mnt2 is the following. The polynomials P_{k}(q) are given by P_{0}(q) = 0, P_{1}(q) = 1 and the recursion formula

P_{k+1}(q) = (2 + q)P_{k}(q) − P_{k−1}(q) .

The numbers B(k, m) of the modified numerical triangle Mnt2 are defined as

B(k, m) = [q^{m}]P_{k}(q) .

Theorem 2.

B(k, m) = \binom{k + m}{2m + 1} .

The paper gives a backward pointer to [2], which also has Mnt1, but also another one, g(k, m), which is a little bit clumsy to describe, but for which I found the formula Theorem 3.

g(k, m) = \frac{k^{2} − k + m^{2}}{m(2m − 1)} \binom{k + m − 2}{2m − 2} .

Date: May 10, 1996.

The paper [1] is also referenced, but it does not give more triangles than the ones just described.

We only give a proof of Theorem 1, the other ones being similar. The formula is evidently true for m = 0 and also for k = 0 and k = 1. Now, proceeding by induction and reading off the coefficient of q^{m} in equation (1) we get for m, k ≥ 1

A(k + 1, m) = 2A(k, m) + A(k, m − 1) − A(k − 1, m) .

It is a routine matter to check that A(k, m) = \binom{k+m}{2m} fulfills this.

References

[1] Z. W. Trzaska. Numerical triangle, Fibonacci sequence and ladder networks: some further results. Appl. Math. Lett., 6.4:55–61, 1993.

[2] Z. W. Trzaska. Modified numerical triangle and the Fibonacci sequence. The Fibonacci Quarterly, 32:124–129, 1994.

[3] Z. W. Trzaska. On Fibonacci hyperbolic trigonometry and modified numerical triangles. The Fibonacci Quarterly, 34:129–138, 1996.

1991 Mathematics Subject Classification. 05A10.