Abstract The aim of this article is to investigate two new classes quaternions, namely, balancing and Lucas-balancing quaternions that are based on numbers, respectively. Further, some identities including Binet’s formulas, summation Catalan’s identity, etc. concerning these also established.

In this work, we define higher-order Jacobsthal–Lucas quaternions with the help of numbers. We examine some identities quaternions. introduce their basic definitions and properties. give Binet’s formula, Cassini’s identity, Catalan’s d’Ocagne generating functions, exponential functions also relations between Jacobsthal

In this paper, with the help of finite operators and Fibonacci numbers, we define a new family quaternions whose components are operator numbers. We also provide some properties these types quaternions. Moreover, derive many identities related to by using matrix representations.

In this study, we define Pauli–Leonardo quaternions by taking the coefficients of Pauli as Leonardo numbers. We give recurrence relation, Binet formula, generating function, exponential some special equalities, and sum properties these novel quaternions. addition, investigate interrelations between Pauli–Fibonacci, Pauli–Lucas Moreover, create algorithms that determine terms Finally, generate m...

The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared for its importance with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.-Clerk Maxwell, 1869. Quaternions came from Hamil...

The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomorphism). The other one, D4, can be constructed as a semi-direct product: D4 ∼= Aff(Z/(4)) ∼= Z/(4) o (Z/(4))× ∼= Z/(4) o Z/(2), where the elements of Z/(2) act on Z/(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We wil...

This paper presents an extended Kalman filter for real-time estimation of rigid body orientation using the newly developed MARG (Magnetic, Angular Rate, and Gravity) sensors. Each MARG sensor contains a three-axis magnetometer, a three-axis angular rate sensor, and a three-axis accelerometer. The filter represents rotations using quaternions rather than Euler angles, which eliminates the long-s...