On the Fibonacci quaternion sequence with quadruple-produce components

This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.

Some properties of Padovan quaternion

In this study, we consider the Padovan Quaternions. Then we investigate the some new properties such as the summation formulas and binomial sum for these quaternions.

Upper estimate of a combinatorial sum

We give an upper estimate for the binomial sum $\begin{array}{} U_{d} \left(x\right)=\sum _{r\ge 0}\binom {d-r+1} r\cdot x^{r} \end{array} $ with natural d and real nonnegative x. In particular, this estimate implies that $\begin{array}{} U_{d} \left(x\right)=O\bigl((0.5+\sqrt{x+0.25} )^{d} \bigr) \end{array} $ with fixed x > 0 and d → ∞.