Convolutions for Pell Polynomials

Pell and Pell–Lucas Numbers with Applications

“…The featured solution is based on the one by Seiffert [214] and the identity P .nC2/aCb D 2Q a P .nC1/aCb C P naCb : (14.21) This follows from the polynomial identity p .nC2/aCb .x/ D p .nC1/aCb .x/ C . 1/ a 1 p naCb .x/; discovered by Horadam and Mahon in 1985, when a is odd [108]. shortly.)…”

“…used to generate Pell and Pell-Lucas polynomials, and we can use it to establish a number of properties of both families, just as Horadam and Mahon did[108].Using induction, we can show that jP n j D jP j n D . 1/ n :This yields the Cassini-like formula for p n .…”

Dual Pell Quaternions

“…and in 1985, Horadam and Mohan obtained Cassini-like formula as follows 8 q n+1 q n1  q 2 n = 8 (1) n+1 .…”

Some results on convolved (p,q)-Fibonacci polynomials

“…[14] In addition, the polynomials u (r) n (x) related to p(x) = 1 and q(x) = x were studied by Dilcher, [12] and those related to p(x) = 2x and q(x) = 1 are in fact the convolutions of Pell polynomials and were expounded in Horadam and Mahon's papers. [16,17] The purpose of this paper is to investigate some properties of the convolved (p, q)-Fibonacci polynomials u (r) n (x). In Section 2, we establish some explicit expressions, recurrence relations and differential recurrence relations for the polynomials u (r) n (x).…”