The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a dierent type of binomial transform.

We investigate the (p, q)−Fibonacci and Lucas octonion polynomials. The main purpose of this paper is using of some properties of the (p, q)−Fibonacci and Lucas polynomials. Also for present some results involving these octonion polynomials, we obtain some interesting computational formulas.

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a different type of binomial transform.

In this work, we introduce bivariate Fibonacci quaternion polynomials andbivariate Lucas quaternion polynomials. We present generating function,Binet formula, matrix representation, binomial formulas and some basicidentities for the bivariate Fibonacci and Lucas quaternion polynomialsequences. Moreover we give various kinds of sums for these quaternionpolynomials.

In this work, we consider generating functions which are generalized tribonacci polynomials Tn(x) and generalized tricobsthal polynomials Jn(x) which are defined in [7]. We derive generating functions for (m + n) − th order of generalized tribonacci polynomials and generalized tricobsthal polynomials for m ≥ 2. Furthermore, we obtain various families of bilinear and bilateral generating functions and give their special cases for these polynomials. Also, we obtain the summation formula of generalized tribonacci polynomials and generalized tricobsthal polynomials.