In this paper, we derive a single formula for the entries of the rth (r ∈ N) power of a certain real circulant matrix of odd and even order, in terms of the Chebyshev polynomials of the first and second kind. In addition, we give two Maple 13 procedures along with some numerical examples in order to verify our calculation.

We study generating functions for the number of permutations in Sn subject to two restrictions. One of the restrictions belongs to S3, while the other to Sk. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second kind. 2000 Mathematics Subject Classification: Primary 05A05, 05A15; Secondary 30B70, 42C05

We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. 2000 Mathematics Subject Classification: Primary 05A05, 05A15; Secondary 30B70, 42C05

We study generating functions for the number of permutations in Sn subject to two restrictions. One of the restrictions belongs to S3, while the other to Sk. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second kind. 2000 Mathematics Subject Classification: Primary 05A05, 05A15; Secondary 30B70, 42C05

We investigate amenability and weak amenability of the l1-algebra of polynomial hypergroups. We derive conditions for (weak) amenability adapted to polynomial hypergroups and show that these conditions are often not satisfied. However, for the hypergroup induced by Chebyshev polynomials of the first kind we prove amenability.

We study generating functions for the number of permutations in Sn subject to two restrictions. One of the restrictions belongs to S3, while the other belongs to Sk. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second kind.

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. 2000 Mathematics Subject Classification: Primary 05A05, 05A15; Secondary 30B70, 42C05

We study generating functions for the number of permutations in Sn subject to set of restrictions. One of the restrictions belongs to S3, while the others to Sk. It turns out that in a large variety of cases the answer can be expressed via continued fractions, and Chebyshev polynomials of the second kind. 2001 Mathematics Subject Classification: Primary 05A05, 05A15; Secondary 30B70 42C05

In this paper, we firstly present a general expression for the entries of the th r   N r power of certain -square n are complex tridiagonal matrix, in terms of the Chebyshev polynomials of the first kind. Secondly, we obtain two complex factorizations for Fibonacci and Pell numbers. We also give some Maple 13 procedures in order to verify our calculations.

We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. 2000 Mathematics Subject Classification: Primary 05A05, 05A15; Secondary 30B70, 42C05