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On a Class of Polynomials with Integer Coefficients
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Milan Janjić

Department of Mathematics and Informatics

University of Banja Luka

Republic of Srpska, Bosnia and Herzegovina

**Abstract:**

We define a certain class of polynomials denoted by
*P*_{n,m,p}(*x*), and give the
combinatorial meaning of the coefficients. Chebyshev
polynomials are special cases of *P*_{n,m,p}(*x*).
We first show
that *P*_{n,m,p}(*x*)
can be expressed in terms of *P*_{n,0,p}(*x*).
From this we derive that *P*_{n,2,2}(*x*)
can be obtained in terms of
trigonometric functions, from which we obtain some of its
important properties. Some questions about orthogonality are also addressed.
Furthermore,
it is shown that *P*_{n,2,2}(*x*)
fulfills the same three-term recurrence
as the Chebyshev polynomials. We also obtain
some other recurrences for
*P*_{n,m,p}(*x*) and its coefficients.
Finally, we derive
a formula for the coefficients of Chebyshev polynomials of the second kind.

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(Concerned with sequences
A001844
A024623
A035597
A049611
A055585
A136388
A136389
A136390
A136397 and
A136398
.)

Received April 12 2008;
revised version received October 29 2008; December 7 2008.
Published in *Journal of Integer Sequences*, December 11 2008.

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