On multivariate orthogonal polynomials and elementary symmetric functions Bracciali, Cleonice F. Piñar González, Miguel Ángel Acknowledgements The authors would like to express their gratitude to the two anonymous reviewers for their useful comments and suggestions, which improved the comprehension of the manuscript. In particular, we thank the reviewer who pointed out references [4–6, 15]. Funding for open access charge: Universidad de Granada / CBUA This research was supported through the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES), in the scope of the CAPES-PrInt Program, process number 88887.310463/2018-00, International Cooperation Project number 88887.468471/2019-00. The second author (MAP) has been partially supported by grant PGC2018-094932-B-I00 from FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación, and the IMAG-María de Maeztu grant CEX2020-001105-M/ AEI/10.13039/501100011033. We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi−xj|2γ+1,x∈(a,b)d, for γ>−1 , where ω(t) is an univariate weight function in t∈(a,b) and x=(x1,x2,…,xd) with xi∈(a,b). Applying the change of variables xi, i=1,2,…,d, into ur, r=1,2,…,d, where ur is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having xi, i=1,2,…,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables ur for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables. 2022-11-30T08:03:18Z 2022-11-30T08:03:18Z 2022-11-01 journal article Bracciali, C.F., Piñar, M.A. On multivariate orthogonal polynomials and elementary symmetric functions. Numer Algor (2022). [https://doi.org/10.1007/s11075-022-01434-4] https://hdl.handle.net/10481/78196 10.1007/s11075-022-01434-4 eng http://creativecommons.org/licenses/by/4.0/ open access Atribución 4.0 Internacional Springer Nature