Factoring Variants of Chebyshev Polynomials of the First and Second Kinds with Minimal Polynomials of cos (2πd)

D. A. Wolfram
2009 ◽  
Vol 19 (09) ◽  
pp. 3005-3016 ◽  

In this paper certain identities are generated for minimal polynomials Θn(z) of numbers 2 cos (2π/n) and for Chebyshev polynomials of the first kind. Against the background of these identities, sets of cyclic points of some modified versions of these polynomials are derived. The discussion leads to the analysis of a certain new counting sequence of numbers connected with Carmichael function.

2010 ◽  
Vol 59 (10) ◽  
pp. 1392-1401 ◽  
Xiaofeng Liao ◽  
Fei Chen ◽  
Kwok-wo Wong

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 74
Waleed Mohamed Abd-Elhameed ◽  
Afnan Ali

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

Bernhard Heim ◽  
Markus Neuhauser

AbstractIn this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and $$0<h(n) \le h(n+1)$$ 0 < h ( n ) ≤ h ( n + 1 ) . We put $$P_0^{g,h}(x)=1$$ P 0 g , h ( x ) = 1 and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ P n g , h ( x ) : = x h ( n ) ∑ k = 1 n g ( k ) P n - k g , h ( x ) . As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind $$\eta $$ η -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

2021 ◽  
Vol 2021 (1) ◽  
Waleed M. Abd-Elhameed ◽  
Youssri H. Youssri

AbstractThe principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

2021 ◽  
pp. 105561
Klaus Schiefermayr ◽  
Maxim Zinchenko

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