On certain bihypernomials related to Pell and Pell-Lucas numbers

Abstract

The bihyperbolic numbers are extension of hyperbolic numbers to four dimensions. In this paper we introduce the concept of Pell and Pell-Lucas bihypernomials as a generalization of bihyperbolic Pell and Pell-Lucas numbers, respectively.

On Bihypernomials Related to Balancing and Chebyshev Polynomials

“…The authors introduced and studied the Fibonacci and Lucas bihypernomials as a generalization of bihyperbolic numbers. In [22], we can find properties of Pell and Pell-Lucas bihypernomials. Bihyperbolic numbers of the Fibonacci type (among others Fibonacci, Pell and Pell-Lucas bihyperbolic numbers) were examined in [6,7].…”

“…. + BhB n (x) = B 0 (x) + B 1 (x)j 1 + B 2 (x)j 2 + B 3 (x)j (x) + B 2 (x)j 1 + B 3 (x)j 2 + B 4 (x)j 3 + • • • + B n (x) + B n+1 (x)j 1 + B n+2 (x)j 2 + B n+3 (x)j B 0 (x) + B 1 (x) + • • • + B n (x) + (B 1 (x) + B 2 (x) + • • • + B n+1 (x) + B 0 (x) − B 0 (x))j 1 + (B 2 (x) + B 3 (x) + • • • + B n+2 (x) + B 0 (x) + B 1 (x) − B 0 (x) − B 1 (x))j 2 + (B 3 (x) + B 4 (x) + • • • + B n+3 (x) + B 0 (x) + B 1 (x) + B 2 (x) − B 0 (x) − B 1 (x) − B 2 (x))j 3 .By(22) we haven l=0 B l (x) = B n+1 (x) − B n (x) − 1 6x − 2 .…”