A chain rule for differentiation with applications to multivariate hermite polynomials

C. S. Withers

doi:10.1017/S0004972700001933

Abstract

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A chain rule is given for differentiating a multivariate function of a multivariate function. In the univariate case this chain rule reduces to Faa de Bruno's formula.

Using this, a simple procedure is given to obtain the rth order multivariate Hermite polynomial from the rth order univariate Hermite polynomial.

References

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Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Withers, C.S. 1985. The moments of the multivariate normal. Bulletin of the Australian Mathematical Society, Vol. 32, Issue. 1, p. 103.

Withers, C.S. 2000. A simple expression for the multivariate Hermite polynomials. Statistics & Probability Letters, Vol. 47, Issue. 2, p. 165.

Leipnik, Roy B. and Pearce, Charles E. M. 2007. The multivariate Faà di Bruno formula and multivariate Taylor expansions with explicit integral remainder term. The ANZIAM Journal, Vol. 48, Issue. 3, p. 327.

Withers, Christopher S. and Nadarajah, Saralees 2008. Multivariate Bell polynomials and their applications to powers and fractionary iterates of vector power series and to partial derivatives of composite vector functions. Applied Mathematics and Computation, Vol. 206, Issue. 2, p. 997.

Kakizawa, Yoshihide and Iwashita, Toshiya 2008. Hotelling's one-sample and two-sample T2 tests and the multivariate Behrens–Fisher problem under nonnormality. Journal of Statistical Planning and Inference, Vol. 138, Issue. 11, p. 3379.

Kakizawa, Yoshihide and Iwashita, Toshiya 2008. A comparison of higher-order local powers of a class of one-way MANOVA tests under general distributions. Journal of Multivariate Analysis, Vol. 99, Issue. 6, p. 1128.

Withers, Christopher S. and Nadarajah, Saralees 2009. Moments from cumulants and vice versa. International Journal of Mathematical Education in Science and Technology, Vol. 40, Issue. 6, p. 842.

Withers, Christopher S. and Nadarajah, Saralees 2011. Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal. ESAIM: Probability and Statistics, Vol. 15, Issue. , p. 340.

Withers, Christopher S. and Nadarajah, Saralees 2011. Estimates of low bias for the multivariate normal. Statistics & Probability Letters, Vol. 81, Issue. 11, p. 1635.

Withers, Christopher S. and Nadarajah, Saralees 2013. Expansions for the distribution of asymptotically chi-square statistics. Statistical Methodology, Vol. 12, Issue. , p. 16.

Terdik, György 2021. Multivariate Statistical Methods. p. 183.

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A chain rule for differentiation with applications to multivariate hermite polynomials

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