Chebyshev polynomials and nested square roots

A complete view of Viète-like infinite products with Fibonacci and Lucas numbers

“…For this to be done, the key tool is our result [2,Theorem 4], which can be considered as a strengthening of the main results in [6,9], and which we give here without proof: …”

“…The basic ingredients for deriving this result were new infinite products of cosines in conjunction with a formula that links nested square roots of 2 with certain cosines (see [1,2]). The main result in [4] reads ia n ffiffiffi 3 p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À b n p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À b n p q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 À ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 À b n p q r Á Á Á ; ð1:3Þ…”

A new estimate for a quantity involving the Chebyshev polynomials of the first kind

“…The celebrated Chebyshev polynomials have a wide range of applications in many fields such as numerical analysis, differential equations, approximation theory, and number theory (see, e.g., [2,9,6]). Some extensions and recent developments of the Chebyshev polynomials can be found, e.g., in [17,8,5,12,11,7,10]. As is well known, the Chebyshev polynomials of the first kind are defined by the recurrence relation T 0 (t) = 1, T 1 (t) = t, T k+1 (t) = 2tT k (t) − T k−1 (t) k = 1, 2, .…”

Differential equation description and Chebyshev approximation of linear time‐invariant circuits

“…22 An n order CP has n different simple roots in the interval from −1 to 1. These Chebyshev roots 23 are known as Chebyshev nodes 24 because the roots are used as the nodes in the polynomial interpolation. 25 Chebyshev polynomials and the related mathematical methods have recently been studied to simulate electronics circuit systems.…”