Abstract
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,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.
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