On a qq-analogue of Faà di Bruno’s determinant formula

Faà di Bruno’s formula is the higher chain rule for differentiation. By means of Gessel’s qq-composition we derive a qq-analogue of Faà di Bruno’s determinant formula for the nnth derivative of a composite function. The formula is regarded as a new form of the qq-analogue of Faà di Bruno’s formula. We also derive qq-analogues of the complete Bell polynomials, which are in the form of a determinant. The qq-complete Bell polynomials include the classical complete Bell polynomials as a special case.