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 φ(i), (i = 1, 2, ..., m), where φ(i) are composite functions of different orders, i.e. φ(i) (t) = (i,1) ((i,2) (... ((i,ri) (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. © 2005 Elsevier Ltd. All rights reserved.
Bernardini, A., Natalini, P., & Ricci, P. E. (2005). Multidimensional bell polynomials of higher order. Computers and Mathematics with Applications, 50(10–12), 1697–1708. https://doi.org/10.1016/j.camwa.2005.05.008