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.

A., B., Natalini, P., P. E., R. (2005). Multi-dimensional Bell polinomials of higher order. COMPUTERS & MATHEMATICS WITH APPLICATIONS, 50, 1697-1708 [10.1016/j.camwa.2005.05.008].

### Multi-dimensional Bell polinomials of higher order

#### 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,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.
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2005
A., B., Natalini, P., P. E., R. (2005). Multi-dimensional Bell polinomials of higher order. COMPUTERS & MATHEMATICS WITH APPLICATIONS, 50, 1697-1708 [10.1016/j.camwa.2005.05.008].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11590/151083`
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