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On a class of polynomials with integer coefficients. (English) Zbl 1196.11050

Summary: We define a certain class of polynomials denoted by \[ P_{n,m,p}(x)=\sum_{i=0}^m{n\choose i}x^{m-i}P_{n-m-i,0,p} (x) \] , 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.

MSC:

11C08 Polynomials in number theory
11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
11B37 Recurrences
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

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