Hosoya-Pell cyclic & proximity groups. application of HP sinusoidal polynomials in modulating oscillatory waveforms emanating from vibrating systems

This paper introduces the periodicity (repeating subsequences) of Hosoya-type Pell orbits as a source of important results for the generation of cyclic groups. Application of time-constrained Hosoya-Pell sinusoidal polynomials $ \boxed{HP_{sp}(t)} $ (introduced in this paper) is given in terms of modulating motion waveforms emanating from vibrating systems. Methods This paper demonstrates how periodicity in Hosoya-Pell subsequences are a source of cyclic groups and in Hosoya-Pell sinusoidal polynomials as a means of modulating highly oscillatory waveforms enclosed in Hilbert envelopes, leading to proximal groups, which provide concise representation of modulating motion waveforms emanating from vibrating systems. Results A main result is the concise representation of the periodicity of Hosoya-type Pell orbits in cyclic groups. This application of $ HP_{sp}(t) $ is threefold, namely, (ⅰ) use of Pell triangle vertex numbers as $ HP_{sp}(t) $ coefficients, (ⅱ) modulation (smoothing) of highly oscillatory motion waveforms as a means of diminishing energy consumption in vibrating systems, and (ⅲ) concise proximal group representation of complex HPsp-modulated waveforms.