Di↵erentiation of multivariable composite functions and Bell Polynomials by Silvia Noschese and Paolo E. Ricci Rome University ”La Sapienza” – Italy
Abstract We generalize the Bell polynomials in order to derive an operational tool for the di↵erentiation 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.
1991 Mathematics Subject Classification. 330XX – 65L05 Key words and phrases. Bell Polynomials. Multivariable composite functions. Implicit functions. Cauchy problem for ordinary di↵erential equations.
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1
Introduction
The purpose of the paper is to develop the theory related to the Bell polynomials 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 di↵erentiation 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 di↵erentiation 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.. Finally, in Section 5 we define the general case of the r-variate Bell polynomials, and generalize the results contained in Section 3.
2
Bell Polynomials
Compose the function f (x) with x = (t), assuming that they are defined in suitable intervals of the t and x-axis. If f and are n times di↵erentiable with respect to the independent variables, the relative Bell polynomials are defined in the following way (see e.g. [2]-[4]): Bn (f1 , f2 , . . . , fn ;
1,
2, . . . ,
2
n)
:= Dn f ( (t)),
m
dk f (x)|(x= (t)) dxk
where we set Dm = dtd m ; For example one has:
B1 (f1 ; B2 (f1 , f2 ; B3 (f1 , f2 , f3 ;
1,
2,
=: fk and Dh (t) =:
1)
= f1
1
1,
2)
= f1
2
3)
= f1
3
+ f2
+ f2 (3
h.
2 1 2 1)
+ f3
3 1.
Inductively, we can write: Bn (f1 , f2 , . . . , fn ;
1,
2, . . . ,
n ) :=
n X
An,k ( 1 ,
2, . . . ,
n )fk ,
k=1
where the generic coefficient An,k , for all k = 1, . . . , n, is a polynomial in 1 , 2 , . . . , n , homogeneous of degree k and isobaric of weight n (i.e. it is a linear combination of monomials k11 k22 · · · knn of weight k1 +2k2 +. . .+nkn = n ). The formula of Fa`a di Bruno (proved for example in [3], by using the umbral calculus) gives us an explicit representation for the Bell polynomials: Dn f ( (t)) =
X
n! 1 n fk1 +...+kn [ ]k1 . . . [ ]kn , k ! . . . k ! 1! n! 1 n ⇡(n)
where the sum is extended to all the partitions of the natural n by means of the nonnegative integers k1 , k2 , . . . , kn : n = k1 + 2k2 + . . . + nkn . Moreover, making use of the Leibniz rule for the nth derivative of the product of two functions, it is possible to write a recurrence formula for the Bell polynomials (n 0), initializing with B0 := f1 : Bn+1 (f1 , f2 , . . . , fn+1 ; =
n X
k=0
!
n Bn k (f2 , f3 , . . . , fn k
1, k+1 ;
2, . . . , 1,
n+1 )
2, . . . ,
= n k ) k+1 .
Among the applications of the Bell polynomials, we stress that, by means of these polynomials, it is possible to obtain general expressions of Newton– Girard formulas for the sums of powers of zeros of entire functions (in particular polynomials). 3
Also, in [1] the Bell polynomials are used to construct representation formulas for the moments of systems of orthogonal polynomials generated by a three–term recurrence relation.
3
Bivariate Bell Polynomials
Let us consider now the composition f ( (1) (t), (2) (t)), make all the standard assumptions about domains of definition and di↵erentiability, and set (h+k) fxk yh := ddxh yk f (x, y)|(x= (1) (t),y= (2) (t)) and Dm (i) (t) =: (i) n , i = 1, 2. We can define, in a natural way, the bivariate Bell polynomials, as follows: Bn(2) (f ;
(1)
;
(2)
) := Dn f (
(1)
(2)
(t),
(t)).
(Note that the Bell polynomials Bn (f1 , f2 , . . . , fn ; 1 , 2 , . . . , n ) could also be written from now on as Bn(1) (f ; ), but we prefer to use the classical notation). We observe that Df (
(1)
(t),
(2)
(t)) = fx
(1) 1
(2) 1
+ fy
= B1 (fx ,
(1) 1 )
(2) 1 ).
+ B1 (fy ,
And, setting
B1 (fx ;
B12 (fx ;
(1) 1 )
:= B2 (fx , fx2 ;
(1) 1 ,
(1) 2 ),
B12 (fy ;
(2) 1 )
:= B2 (fy , fy2 ;
(2) 1 ,
(2) 2 ),
(1) 1 )
B1 (fy ;
(2) 1 )
= fx
(1) 1
fy
(2) 1
:= fxy
(1) (2) 1 1 ,
we get D2 f (
(1)
(t),
B2 (fx , fx2 ;
(2)
(t)) = fx2 (
(1) 1 ,
= B12 (fx ;
(1) 2 )
(1) 2 (1) (2) 2 (2) (1) (2) 1 ) + fx 2 + fy 2 ( 1 ) + fy 2 + 2fxy 1 1
+ 2B1 (fx ;
(1) 1 )
+ 2B1 (fx ;
(1) 1 )
B1 (fy ;
(1) 1 )
4
B1 (fy ;
(2) 1 )
+ B2 (fy , fy2 ;
(2) 1 )
+ B12 (fy ;
(2) 1 ,
(2) 1 )
=
(2) 2 )
=
=
(1) 1 )
= [B1 (fx ;
(2) 2 1 )] .
+ B1 (fy ;
In the above formula the operation symbol must be interpreted as the usual product, with the convention that it acts in an operational way in the case of partial derivatives, i.e.:
fxh
(2) 1 )
B1m (fy ;
(and
(1) 1 )
B1m (fx ;
fyk := fxh yk ;
= Bm (fx , . . . , fxm ; (2) 1 ,...,
= B2 (fy , . . . , fym ;
(1) 1 ,...,
(1) m )
(2) m )).
Therefore, we can derive, by induction, the following result. Theorem 3.1 Let f (x, y), (1) (t), polynomial of the system {Bn(2) (f ; following rule: Bn(2) (f ;
(1)
;
(2)
(2) (1)
(t) be analytical functions. The n-th ; (2) )}n2N , can be computed with the (1) 1 )
) = [B1 (fx ;
+ B1 (fy ;
(2) n 1 )] .
By using the Leibniz rule we can write Dn+1 f (
(1)
(t),
=
(2)
(t)) = Dn Df (
n X
k=0
!
n D n k fx k
(1)
(1) k+1
(t),
+
(2)
n X
k=0
(t)) = Dn [fx !
n D n k fy k
(1) 1
+ fy ,
(2) 1 ]
=
(2) k+1 .
It is easy now to obtain a recurrence formula for the bivariate Bell polynomials. Theorem 3.2 (2)
B0 (fx ;
(1)
;
(2)
(2)
=
n X
k=0
!
n (2) B (fx ; k n k
(2)
B0 (fy ;
) := fx ;
(1)
Bn+1 (f ; (1)
;
(2)
)
;
(1) k+1
5
(2)
(1)
;
(2)
) := fy ;
)= (2)
+ Bn k (fy ;
(1)
;
(2)
)
(2) k+1 .
4
Applications
We treat here only the simplest applications to the di↵erentiation of composite functions which appear in several classical topics. The use of the bivariate Bell polynomials allow us to obtain representation formulas for all the involved subsequent derivatives. Let us consider two problems among the most important ones: the implicit function problem and the Cauchy problem for O.D.E..
4.1
Implicit Function Problem
Let f (x, y) 2 C m be a function defined in A⇥B , where A and B are two open intervals of the x and y-axis, and suppose f (x0 , y0 ) = 0 and fy (x0 , y0 ) 6= 0,. The function y = y(x) 2 C m in a suitable subinterval I ✓ A (containing x0 ) is implicitly defined by the equation f (x, y) = 0 if, 8x 2 I, f (x, y(x)) = 0. Moreover classical results state that y(x) is unique and satisfies y 0 (x) =
fx (x, y(x)) , fy (x, y(x))
y(x0 ) = y0 .
Now, in order to find y = y(x) we need to solve such a Cauchy problem (see next subsection). On the other hand we could directly obtain a polynomial approximation of the local solution using the Taylor expansion in x0 and the Bell polynomials.
4.2
Cauchy Problem for O.D.E.
Let us consider an initial value problem for O.D.E.. One of the simplest approach for obtaining the numerical solution is given by the one–step methods in which one defines a discretization of the problem by means of a di↵erence equation of the first order derived from Taylor series. If the function f (x, y) 2 C m in A ⇥ R = {(x, y)|a x b; y 2 R}, the solution y(x) can be theoretically expanded in a suitable neighborhood of a generic x 2 [a, b], (x + h 2 [a, b]) : 6
hm (m) y (x) + O(hm+1 ). m! This is one of the possible applications of the bivariate Bell polynomials. Define (1) (t) := t and (2) (t) := y(t). Require that the local truncation error, i.e. the error for individual steps, is of order hm+1 . Use the Bell polynomials for m 1 terms of the expansion. y(x + h) = y(x) + hy 0 (x) + . . . +
y 0 (t) = f (t, y(t)), y 00 (t) = B1 (fx , 1) + B1 (fy , y 0 ), y 000 (t) = [B1 (fx , 1) + B1 (fy , y 0 )]2 , ... y (m) (t) = [B1 (fx , 1) + B1 (fy , y 0 )]m m
=
+
0
m
+
1 1
m
1 2
+
!
1
!
!
1
=
Bm 1 (fx , fx2 , . . . , fxm 1 ; 1, 0, . . . , 0)
Bm 2 (fx , fx2 , . . . , fxm 2 ; 1, 0, . . . , 0) B1 (fy ; y 0 )
Bm 3 (fx , fx2 , . . . , fxm 3 ; 1, 0, . . . , 0) B2 (fy , fy2 ; y 0 , y 00 )
m m
!
+...
1 Bm 1 (fy , fy2 , . . . , fym 1 ; y 0 , y 00 , . . . , y (m 1
1)
).
Note that the computation of the Bell polynomials appearing in the last formula can be performed by induction or by using the Fa`a di Bruno formula.
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5
r-variate Bell polynomials
We compose the (sufficiently) regular function f (x1 , . . . , xr ) with r regular functions (i) (t), i = 1, . . . , r and, assuming verified all the standard conditions. Then considerations of the preceding section can be extended in an analogous way by defining the r-variate Bell Polynomials as follows: Bn(r) (f ;
(1)
(r)
;...;
) := Dn f (
(1)
(t), . . . ,
(r)
(t)).
Observe that Df (
(1)
(r)
(t), . . . ,
(1) 1 +. . .+fxr
(t)) = fx1
(r) 1
= B1 (fx1 ,
(1) 1 )+. . .+B1 (fxr ,
(r) 1 ),
and, developing the second derivative of a composite function: D2 f ( [fx21
(1) 1
+ f x1 x2
+[fx2 x1
(1) 1
+ fx22
+[fxr x1
(1) 1
(2) 1
(2) 1
(1)
(r)
(t), . . . ,
(r) (1) 1 ] 1
+ . . . + f x1 xr
+ . . . + f x2 xr
+ f xr x2
(t)) =
(2) 1
(r) (2) 1 ] 1
+ . . . + fx2r
+ f x1
+ f x2
(r) (r) 1 ] 1
(1) 2 +
(2) 2
+ ...+
+ f xr
(r) 2 ,
we get a result analogous to the preceding case: D2 f (
(1)
(t), . . . ,
(r)
(t)) = [B1 (fx1 ;
(1) 1 )
+ . . . + B1 (fxr ;
(r) 2 1 )] .
Now, proceeding inductively, we obtain the following generalization of Theorem 3.1. Theorem 5.1 Bn(r) (f ;
(1)
;...;
(r)
) = [B1 (fx1 ;
(1) 1 )
+ . . . + B1 (fxr ;
(r) n 1 )] .
The analogue of Theorem 3.2 is the following recurrence formula for the r-variate Bell polynomials.
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Theorem 5.2 (r)
B0 (fxj ;
(1)
;...;
(r)
=
n X r X
k=0 j=1
Bn+1 (f ;
(r)
) := fxj
(1)
;...;
!
n (r) B (fx ; k n k j
(1)
(r)
j = 1, . . . , r; )=
;...;
(r)
)
(j) k+1 .
References [1] B. GERMANO - P.E. RICCI: Representation formulas for the moments of the density of zeros of orthogonal polynomial sets, Le Matematiche, XLVIII (1993), 77–86. [2] J. RIORDAN: An Introduction to Combinatorial Analysis, J. Wiley & Sons, New York (1958). [3] S.M. ROMAN: The Fa` a di Bruno Formula, Amer. Math. Monthly, 87 (1980), 805–809. [4] S.M. ROMAN - G.C. ROTA: The umbral calculus, Advances in Math., 27 (1978), 95–188.
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Address: Silvia NOSCHESE Dipartimento di Matematica “Guido CASTELNUOVO” Universit`a degli Studi di Roma “La Sapienza”, Italy Email:
[email protected] Paolo Emilio RICCI Dipartimento di Matematica “Guido CASTELNUOVO” Universit`a degli Studi di Roma “La Sapienza”, Italy Email:
[email protected] Fax #: (39) - 0644701007
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