On Six consecutive pairs of Lucas and Fibonacci-type $p$ entities connected to Chebyshev polynomials

R.  Rangarajan; C. Goutham; K. Bharatha; P. Siva Kota  Reddy

doi:10.5269/bspm.76726

R. Rangarajan https://orcid.org/0000-0002-0305-7761
C. Goutham https://orcid.org/0000-0002-0305-7761
K. Bharatha ST PHILOMENA'S COLLEGE, MYSORE https://orcid.org/0009-0000-1459-5699
P. Siva Kota Reddy

Abstract

Let p be a chosen positive integer parameter. Let Np = (p+1)2 +1 and Mp = 2(p+1)2 +1. We denote the p-entities by {(λ6n+k, θ6n+k) : k = 0, 1, 2, 3, 4, 5; n = 0, 1, 2, 3, · · · }. When p = 1 they are equal to (L6n+k, F6n+k) where Ln and Fn are the well known Lucas and Fibonacci numbers. Also, if x = λ6n+k and y = θ6n+k, they satisfy generalised Pell’s equation x2 − Npy2 = ±(p + 1)2. In the present paper, it will be shown that λ6n+k and θ6n+k are expressible in terms of Tn(Mp) and Un−1(Mp) where, Tn(x) and Un−1(x) are well known Chebyshev polynomials of the first and the second kind. Some interesting combinatorial identities are also derived for each p− entities.

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