  We introduce B-splines on the line of quaternionic order $B_q$ ($q$ in the
algebra of quaternions) for the purposes of multi-channel signal and image
analysis. The functions $B_q$ are defined first by their Fourier transforms,
then as the solutions of distributional differential equation of quaternionic
order. The equivalence of these definitions requires properties of quaternionic
Gamma functions and binomial expansions, both of which we investigate. The
relationship between $B_q$ and a backwards difference operator is shown,
leading to a recurrence formula. We show that the collection of integer shifts
of $B_q$ is a Riesz basis for its span, hence generating a multiresolution
analysis. Finally, we demonstrate the pointwise and $L^p$ convergence of the
quaternionic B-splines to quarternionic Gaussian functions.
