This paper presents a new kind of splines, called non-uniform algebraic-trigonometric T-splines (NUAT T-splines) of odd degree, and also gives a local refinement algorithm for this kind of splines. Similar to the definition of the T-splines, the NUAT T-splines are defined by applying the T-spline framework to the non-uniform algebraic-trigonometric B-splines (NUAT B-splines). Based on the knot insertion algorithm of the NUAT B-splines, a local refinement algorithm for odd degree NUAT T-splines is showed. This algorithm guarantees that the resulting control grid is a T-mesh like the original one. Finally, this paper proves that, for any odd degree NUAT T-spline, the linear independence of its blending functions can be determined by computing the rank of the NUAT T-spline-to-NUAT B-spline transform matrix.