Differential operators associated with zonal polynomials. II

Donald St P. Richards

doi:10.1007/BF02481013

Differential operators associated with zonal polynomials. II

Donald St P. Richards

Statistics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Let C_κ(S) be the zonal polynomial of the symmetric m×m matrix S=(s_ij), corresponding to the partition κ of the non-negative integer k. If ∂/∂S is the m×m matrix of differential operators with (i, j)th entry ((1+δ_ij)∂/∂s_ij)/2, δ being Kronecker's delta, we show that C_k(∂/∂S)C_λ(S)=k!δ_λkC_k(I), where λ is a partition of k. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.

Original language	English (US)
Pages (from-to)	119-121
Number of pages	3
Journal	Annals of the Institute of Statistical Mathematics
Volume	34
Issue number	1
DOIs	https://doi.org/10.1007/BF02481013
State	Published - Dec 1982

All Science Journal Classification (ASJC) codes

Statistics and Probability

Access to Document

10.1007/BF02481013

Cite this

@article{c31f6d8503244d70929c882bda73a0c7,

title = "Differential operators associated with zonal polynomials. II",

abstract = "Let C κ(S) be the zonal polynomial of the symmetric m×m matrix S=(sij), corresponding to the partition κ of the non-negative integer k. If ∂/∂S is the m×m matrix of differential operators with (i, j)th entry ((1+δij)∂/∂sij)/2, δ being Kronecker's delta, we show that Ck(∂/∂S)Cλ(S)=k!δλkCk(I), where λ is a partition of k. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.",

author = "Richards, \{Donald St P.\}",

year = "1982",

month = dec,

doi = "10.1007/BF02481013",

language = "English (US)",

volume = "34",

pages = "119--121",

journal = "Annals of the Institute of Statistical Mathematics",

issn = "0020-3157",

publisher = "Springer Netherlands",

number = "1",

}

TY - JOUR

T1 - Differential operators associated with zonal polynomials. II

AU - Richards, Donald St P.

PY - 1982/12

Y1 - 1982/12

N2 - Let C κ(S) be the zonal polynomial of the symmetric m×m matrix S=(sij), corresponding to the partition κ of the non-negative integer k. If ∂/∂S is the m×m matrix of differential operators with (i, j)th entry ((1+δij)∂/∂sij)/2, δ being Kronecker's delta, we show that Ck(∂/∂S)Cλ(S)=k!δλkCk(I), where λ is a partition of k. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.

AB - Let C κ(S) be the zonal polynomial of the symmetric m×m matrix S=(sij), corresponding to the partition κ of the non-negative integer k. If ∂/∂S is the m×m matrix of differential operators with (i, j)th entry ((1+δij)∂/∂sij)/2, δ being Kronecker's delta, we show that Ck(∂/∂S)Cλ(S)=k!δλkCk(I), where λ is a partition of k. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.

UR - https://www.scopus.com/pages/publications/51649164412

UR - https://www.scopus.com/pages/publications/51649164412#tab=citedBy

U2 - 10.1007/BF02481013

DO - 10.1007/BF02481013

M3 - Article

AN - SCOPUS:51649164412

SN - 0020-3157

VL - 34

SP - 119

EP - 121

JO - Annals of the Institute of Statistical Mathematics

JF - Annals of the Institute of Statistical Mathematics

IS - 1

ER -

Differential operators associated with zonal polynomials. II

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this