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DOI :10.26650/ijmath.2024.00011   IUP :10.26650/ijmath.2024.00011    Full Text (PDF)

On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument

Hironobu Kimura

We investigate the several special functions defined by a matrix integral on the Hermitian matrix space of size n. They are the matrix argument analogues of the Gauss hypergeometric, Kummer’s confluent hypergeometric, the Bessel, the Hermite-Weber and Airy functions which play important roles in the multivariate statistical analysis and the random matrix theory. We give the integral representations for them as functions of eigenvalues of the matrix argument by using the result of Harish-Chandra and Itzykson-Zuber, and give the systems of differential equations for them. We show that these system are holonomic and have the holonomic rank 2 𝑛 using the theory of Gröbner basis.

Keywords: hypergeometric function, matrix integral, holonomic system, Gröbner basis

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APA

Kimura, H. (2024). On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument. Istanbul Journal of Mathematics, 2(1), 1-27. https://doi.org/10.26650/ijmath.2024.00011


AMA

Kimura H. On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument. Istanbul Journal of Mathematics. 2024;2(1):1-27. https://doi.org/10.26650/ijmath.2024.00011


ABNT

Kimura, H. On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument. Istanbul Journal of Mathematics, [Publisher Location], v. 2, n. 1, p. 1-27, 2024.


Chicago: Author-Date Style

Kimura, Hironobu,. 2024. “On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument.” Istanbul Journal of Mathematics 2, no. 1: 1-27. https://doi.org/10.26650/ijmath.2024.00011


Chicago: Humanities Style

Kimura, Hironobu,. “On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument.” Istanbul Journal of Mathematics 2, no. 1 (Oct. 2024): 1-27. https://doi.org/10.26650/ijmath.2024.00011


Harvard: Australian Style

Kimura, H 2024, 'On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument', Istanbul Journal of Mathematics, vol. 2, no. 1, pp. 1-27, viewed 11 Oct. 2024, https://doi.org/10.26650/ijmath.2024.00011


Harvard: Author-Date Style

Kimura, H. (2024) ‘On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument’, Istanbul Journal of Mathematics, 2(1), pp. 1-27. https://doi.org/10.26650/ijmath.2024.00011 (11 Oct. 2024).


MLA

Kimura, Hironobu,. “On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument.” Istanbul Journal of Mathematics, vol. 2, no. 1, 2024, pp. 1-27. [Database Container], https://doi.org/10.26650/ijmath.2024.00011


Vancouver

Kimura H. On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument. Istanbul Journal of Mathematics [Internet]. 11 Oct. 2024 [cited 11 Oct. 2024];2(1):1-27. Available from: https://doi.org/10.26650/ijmath.2024.00011 doi: 10.26650/ijmath.2024.00011


ISNAD

Kimura, Hironobu. “On the holonomic systems for the Gauss hypergeometric function and its confluent family of a matrix argument”. Istanbul Journal of Mathematics 2/1 (Oct. 2024): 1-27. https://doi.org/10.26650/ijmath.2024.00011



TIMELINE


Submitted08.04.2024
Accepted07.06.2024
Published Online25.06.2024

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