Janjić, Milan On a class of polynomials with integer coefficients. (English) Zbl 1196.11050 J. Integer Seq. 11, No. 5, Article ID 08.5.2, 9 p. (2008). 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.) Keywords:Chebyshev polynomials; binomial coefficients; recurrence relations Software:OEIS PDFBibTeX XMLCite \textit{M. Janjić}, J. Integer Seq. 11, No. 5, Article ID 08.5.2, 9 p. (2008; Zbl 1196.11050) Full Text: arXiv EuDML EMIS Online Encyclopedia of Integer Sequences: Centered square numbers: a(n) = 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z; then sequence gives Z values. a(n) = T(n,2), array T as in A049600. Second column of triangle A055584. Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,2}(x) with 0 omitted (exponents in increasing order). Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,3}(x) with 0 omitted (exponents in increasing order). Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,4}(x) with 0 omitted (exponents in increasing order). Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,5}(x) with 0 omitted (exponents in increasing order). Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,6}(x) with 0 omitted (exponents in increasing order).