gives the partial Bell polynomial .
gives the generalized partial Bell polynomial of a matrix m.
gives the generalized Bell polynomial of a matrix m.
- Mathematical function, suitable for both symbolic and numeric manipulations.
- The BellY polynomial is given by ⋯ Boole[m1+2 m2+⋯+n mnn∧m1+m2+⋯+mnk] (xs/s!)ms.
- The partial Bell polynomial can be used to express the derivative of a composition of two functions through the Faà di Bruno formula .
- The generalized Bell polynomial can be used to express the derivative of a composition of functions .
- BellY[n,k,m] is equivalent to BellY where is formed by adding UnitVector[n,k] to m as a first column.
Examplesopen allclose all
Basic Examples (3)
The generalized chain rule allows one to directly compute the derivative of using BellY: . Verify this for symbolic and for low orders of :
From the formula, it can be directly seen that the derivative is linear in with coefficients that are polynomial in , where is the polynomial coefficient of . Define a typesetting that makes this relation be obvious by paneling the Bell coefficients:
Compare with BellY:
Recover BellB[n,z] as a special case: