A note on Mehler's formula

Authors

  • Alessandri Canchoa Quispe Departamento Académico de Matemática, Universidad Nacional Agraria La Molina, Av. La Molina s/n, Lima 12, Perú.
  • Sergio Camiz CNR-ISPC, Istituto di Scienze del Patrimonio Culturale, Piazzale Aldo Moro, 5, 00185 Roma, Italia.
  • Eladio Ocaña IMCA, Instituto de Matemática y Ciencias Afines, Universidad Nacional de Ingeniería, Calle Los Biólogos 245 Urb. San César, La Molina, Lima 12, Perú.

DOI:

https://doi.org/10.17268/sel.mat.2025.01.07

Abstract

In this paper we propose an original proof of Mehler expansion of the Gaussian distribution in terms of probabilistic Hermite polynomials.

References

Hermite M. Sur un nouveau développement en série des fonctions. Imprimerie de Gauthier-Villars; 1864.

Andreani C, Senesi R, Krzystyniak M, Romanelli G, Fernandez-Alonso F. Atomic Quantum Dynamics in Materials Research. In: Fernandez-Alonso F, Price DL, editors. Neutron Scattering - Applications in Biology, Chemistry, and Materials Science. vol. 49 of Experimental Methods in the Physical Sciences. Academic Press; 2017. p.403-57.

Asli BHS, Flusser J. Discrete Hermite Moments and Their Application in Chemometrics. Chemometrics and Intelligent Laboratory Systems. 2018; 177:83-8.

Hall P. The Bootstrap and Edgeworth Expansion. Springer Science & Business Media; 2013.

Baccini A, Caussinus H, de Falguerolles A. Diabolic Horseshoes. In: 9th International Workshop on Statistical Modelling, Exeter. Citeseer; 1994. p. 11-5.

Mehler FG. Uber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen hoherer Ordnung. Journal für die reine und angewandte Mathematik. 1866.

Watson GN. Notes on Generating Functions of Polynomials: (1) Laguerre Polynomials. Journal of the London Mathematical Society. 1933;1(3):189-92.

Kibble WF. An Extension of a Theorem of Mehler’s on Hermite Polynomials. Mathematical Proceedings of the Cambridge Philosophical Society. 1945;41(1):12-5.

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Published

2025-07-26

How to Cite

Canchoa Quispe, A., Camiz, S., & Ocaña, E. (2025). A note on Mehler’s formula. Selecciones Matemáticas, 12(01), 86 - 89. https://doi.org/10.17268/sel.mat.2025.01.07

Issue

Vol. 12 No. 01 (2025): January - July

Section

Articles

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