On a -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 q -composition we derive a q -analogue of Faà di Bruno’s determinant formula for the n th derivative of a composite function. The formula is regarded as a new form of the q -analogue of Faà di Bruno’s formula. We also derive q -analogues of the complete Bell polynomials, which are in the form of a determinant. The q -complete Bell polynomials include the classical complete Bell polynomials as a special case.