A generalized Bell polynomial is a multivariate polynomial that expresses a derivative of a function
composition of any finite number of functions. For
positive integers 
and 
and an 
matrix 
, define the polynomials
 |
(1)
|
The generalized Bell polynomial is
![Y_n(A)=n![t^n]F_1(F_2(...F_m(t)...)),](/images/equations/GeneralizedBellPolynomial/NumberedEquation2.svg) |
(2)
|
where ![[t^n]](/images/equations/GeneralizedBellPolynomial/Inline5.svg)
extracts the coefficient of 
.
For a function composition 
of 
-times differentiable functions,
column 
contains the first 
derivatives of 
, evaluated at 
, and the final column is evaluated at
.
Then 
gives the 
th
derivative of the function
composition. For two columns, it reduces to
 |
(3)
|
where 
is the exponential partial Bell polynomial.
For example,
![Y_2([a_(1,1) a_(1,2); a_(2,1) a_(2,2)])=a_(2,1)a_(1,2)^2+a_(1,1)a_(2,2).](/images/equations/GeneralizedBellPolynomial/NumberedEquation4.svg) |
(4)
|
Generalized Bell polynomials are implemented in the Wolfram Language as BellY[matrix].
