Spinor algebra of k-balancing and k-Lucas-balancing numbers

In this paper, we introduce and study a spinor algebra of [Formula: see text]-balancing numbers referred to as the [Formula: see text]-balancing and [Formula: see text]-Lucas-balancing spinors. First, we give [Formula: see text]-balancing quaternions and their some algebraic properties. Then we introduce a spinor family of [Formula: see text]-balancing numbers by defining a linear and injective correspondence between [Formula: see text]-balancing and [Formula: see text]-Lucas-balancing quaternions to spinors. Here, we give various algebraic properties of this spinor such as the Binet form, Catalan’s identity, d’Ocagne identity, generating and exponential generating functions. Moreover, we obtain various partial sum formulae for these sequences in closed form and also give matrix representations.