In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials ...

1. P. Bateman, J. Kalb, and A. Stenger, A limit involving least common multiples, this MONTHLY 109 (2002) 393–394. doi:10.2307/2695513 2. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel, DordrechtHolland, 1974. 3. B. Farhi, Nontrivial lower bounds for the least common multiple of some finite sequences of integers, J. Number Theory 125 (2007) 393–411. doi:...

The mathematical theory of closed form functions for calculating LSFs on the basis of generating functions is presented. Exploiting recurrence relationships in the series expansion of Chebyshev polynomials of the first kind makes it possible to bootstrap iterative LSF-search from a set of characteristic polynomial zeros. The theoretical analysis is based on decomposition of sequences into symme...

Chebyshev polynomials of the first and the second kind in n variables z. , Zt , ... , z„ are introduced. The variables z, , z-,..... z„ are the characters of the representations of SL(n + 1, C) corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami op...

In this paper we describe the use of multivariate Chebyshev polynomials in computing spectral derivations and Clenshaw–Curtis type quadratures. The multivariate Chebyshev polynomials give a spectrally accurate approximation of smooth multivariate functions. In particular we investigate polynomials derived from the A2 root system. We provide analytic formulas for the gradient and integral of A2 ...

The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which based on the spectral tau method. There are two sets of basis functions used. first set non-symmetric polynomials, namely, shifted Chebyshev polynomials sixth-kind (CPs6), and second modified CPs6. approximation solution written as product chosen function sets. For met...

We study linear transformations $$T :\mathbb {R}[x] \rightarrow \mathbb {R}[x]$$ of the form $$T[x^n]=P_n(x)$$ where $$\{P_n(x)\}$$ is a real orthogonal polynomial system. With $$T=\sum \tfrac{Q_k(x)}{k!}D^k$$ , we seek to understand behavior transformation T by studying roots $$Q_k(x)$$ . prove four main things. First, show that only case are constant and an system when $$P_n(x)$$ shifted set ...

formulae expressing explicitly the coefficients of an expansion of double jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. extension to expansion of triple jacobi polynomials is given. the results for the special cases of double and triple ultraspher...

We show that the resultants with respect to x of certain linear forms in Chebyshev polynomials with argument x are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-r...