Multidimensional bell polynomials of higher order

We develop some extensions of the classical Bell polynomials, previously obtained, by introducing a further class of these polynomials called multidimensional Bell polynomials of higher order. They arise considering the derivatives of functions f in several variables @f^(^i^), (i = 1, 2, ..., m), where @f^(^i^) are composite functions of different orders, i.e. @f^(^i^) (t) = ^(^i^,^1^) (^(^i^,^2^) (... (^(^i^,^r^"^i^) (t))), (i = 1, 2, ..., m). We show that these new polynomials are always expressible in terms of the ordinary Bell polynomials, by means of suitable recurrence relations or formal multinomial expansions. Moreover, we give a recurrence relation for their computation.