Differentiation of Multivariable Composite Functions and Bell Polynomials Silvia Noschese 1 and Paolo E. Ricci 1,2 We generalize the Bell polynomials in order to derive an operational tool for the differentiation of composite functions in several variables. In particular we show a formula that relates the Bell polynomials for multivariable composite functions to the classical ones. Some applications are suggested. KEY WORDS: Bell polynomials; multivariable composite functions; implicit functions; Cauchy problem for ordinary differential equations. 1. INTRODUCTION The purpose of the paper is to develop the theory related to the Bell poly- nomials in order to apply it to classical problems. In Section 2 we describe the main properties of the Bell polynomials: representation formulas and recurrence relation, and briefly recall some of their applications. Section 3 is devoted to the definition of the Bell polynomials related to differentiation of two variables composite functions, (we will refer to them, shortly, as the bivariate Bell polynomials) extending in a natural way the classical Bell polynomials, in order to derive an operational tool for the differentiation of composite functions in two variables. Also, we obtain a formula connecting the bivariate Bell polynomials to the classical ones, and we show a recurrence formula satisfied by them. In Section 4 some possible applications are suggested: in particular we show how one could apply the Bell polynomials and the formula in Theorem 3.1 in order to compute the subsequent derivatives appearing in the well known numerical methods, based on Taylor expansion, approximating the solution of the classical initial value Cauchy problem for O.D.E.. 1 Dipartimento di Matematica ‘‘Guido CASTELNUOVO’’, Universita` degli Studi di Roma ‘‘La Sapienza’’, Italy. 2 To whom correspondence should be addressed. E-mail: Paoloemilio.Ricci@uniromal.it 333 @ 2003 Plenum Publishing Corporation 1521-1398/03/0700-0333/0 Journal of Computational Analysis and Applications, Vol. 5, No. 3, July 2003 (# 2003)