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

• Published in: Discrete mathematics Vol. 311; no. 6; pp. 387 - 392
• Main Authors: Xu, Aimin, Cen, Zhongdi
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 28.03.2011; Elsevier
• Subjects: [formula omitted]-analogue; Algebra; Bells; Chains; Combinatorics; Combinatorics. Ordered structures; Complete Bell polynomial; Composite functions; Derivatives; Determinant; Determinants; Differentiation; Exact sciences and technology; Faà di Bruno’s formula; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Complete Bell polynomial; q -analogue; Faà di Bruno’s formula; Determinant; q-analogue; Composition; Bell polynomial; Faà di Bruno's formula; Discrete mathematics; Function derivative; Combinatorics; Differentiation
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2010.12.001

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.