Journal of Integer Sequences, Vol. 11 (2008), Article 08.5.2

On a Class of Polynomials with Integer Coefficients

Milan Janjić
Department of Mathematics and Informatics
University of Banja Luka
Republic of Srpska, Bosnia and Herzegovina


We define a certain class of polynomials denoted by Pn,m,p(x), and give the combinatorial meaning of the coefficients. Chebyshev polynomials are special cases of Pn,m,p(x). We first show that Pn,m,p(x) can be expressed in terms of Pn,0,p(x). From this we derive that Pn,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 Pn,2,2(x) fulfills the same three-term recurrence as the Chebyshev polynomials. We also obtain some other recurrences for Pn,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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