Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

A Multivariate Faa di Bruno Formula with Applications
HTML articles powered by AMS MathViewer

by G. M. Constantine and T. H. Savits PDF
Trans. Amer. Math. Soc. 348 (1996), 503-520 Request permission

Abstract:

A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of an infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.
References
  • C. S. Chen and T. H. Savits, Some remarks on compound nonhomogeneous Poisson processes, Statist. Probab. Lett. 17 (1993), no. 3, 179–187. MR 1229935, DOI 10.1016/0167-7152(93)90165-F
  • Gregory M. Constantine, Combinatorial theory and statistical design, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987. MR 891185
  • G. M. Constantine and T. H. Savits, A stochastic process interpretation of partition identities, SIAM J. Discrete Math. 7 (1994), no. 2, 194–202. MR 1271991, DOI 10.1137/S0895480192232692
  • Faa di Bruno, C. F. (1855). Note sur une nouvelle formule du calcul differentiel. Quart. J. Math., 1, 359-360.
  • Hoppe, R. (1871). Ueber independente Darstellung der höheren differentialquotienten. Mathematische Annalen, 4, 85-87.
  • Leetsch C. Hsu, Finding some strange identities via Faa di Bruno’s formula, J. Math. Res. Exposition 13 (1993), no. 2, 159–165 (English, with English and Chinese summaries). MR 1223903
  • Lacroix, S. F. (1810). Traité du calcul integral. Tome Premier, Paris.
  • L. Lovász, Combinatorial problems and exercises, North-Holland Publishing Co., Amsterdam-New York, 1979. MR 537284
  • Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
  • Peter McCullagh, Tensor methods in statistics, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1987. MR 907286
  • Most, R. (1871). Ueber die höheren differentialquotienten. Mathematische Annalen, 4, 499-504.
  • Emanuel Parzen, Stochastic processes, Holden-Day Series in Probability and Statistics, Holden-Day, Inc., San Francisco, Calif., 1962. MR 0139192
Similar Articles
Additional Information
  • G. M. Constantine
  • Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
  • Email: gmc@vms.cis.pitt.edu
  • T. H. Savits
  • Affiliation: Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
  • Email: ths@stat.pitt.edu
  • Received by editor(s): January 20, 1994
  • Additional Notes: The first author was funded under a Fulbright grant; the second author was supported by NSF DMS-9203444 and NSA MDA 904-95-H1011
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 503-520
  • MSC (1991): Primary 05A17, 05A19; Secondary 26B05, 60G20
  • DOI: https://doi.org/10.1090/S0002-9947-96-01501-2
  • MathSciNet review: 1325915