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 ...

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...

We develop a neuroevolution-potential (NEP) framework for generating neural network based machine-learning potentials. They are trained using an evolutionary strategy performing large-scale molecular dynamics (MD) simulations. A descriptor of the atomic environment is constructed on Chebyshev and Legendre polynomials. The method implemented in graphic processing units within open-source GPUMD p...

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:...

We present a semi-analytic approach to solving the Boltzmann equation describing the comptonisation of low frequency input photons by a thermal distribution of electrons in the Thomson limit. Our work is based on the formulation of the problem by Titarchuk & Lyubarskij (1995), but extends their treatment by accommodating an arbitrary anisotropy of the source function. To achieve this, we expand...

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...