An analytic way to prove the explicit formula for Hermite polynomial after heat flow deformation and observation in 3D dimensions

Jihao Liu

doi:10.54254/3029-0880/2025.29230

Advances in Operation Research and Production ManagementOpen access

An analytic way to prove the explicit formula for Hermite polynomial after heat flow deformation and observation in 3D dimensions

Research Article

Open Access

An analytic way to prove the explicit formula for Hermite polynomial after heat flow deformation and observation in 3D dimensions

Jihao Liu ^1*

¹ Guangdong Technion-Israel Institute of Technology

^*Corresponding author: jihao.liu@gtiit.edu.cn

Published on 4 November 2025

AORPM Vol.4 Issue 3

ISSN (Print): 3029-0899

ISSN (Online): 3029-0880

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Abstract

Deformation of polynomials is a kind of operation where we add a new variable to the original polynomial. In our case, suppose P is a monic polynomial of degree n with complex coefficients. We evolve P with respect to time by heat flow, creating a function P(t,z) of two variables with given initial dataP(0,z)=P(z)for which∂tP(t,z)=∂zzP(t,z). In this paper, we focus on the deformed polynomial P(t,z). First, we proved the Taylor series representation of deformed polynomial. Then we apply the results to the classical Hermite polynomials and extend to the case of matrix-valued polynomials. From the inspiration of deformed polynomials’ roots movement, we proved the behavior of Hermite polynomials after heat flow deformation and got an explicit formula. For further work, similar to what we have done in this paper, we want to have an explicit formula for deformed matrix Hermite polynomials and give a proof.

Keywords:

math, heat flow, polynomials, zeros, polynomials deformation

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References

[1]. Tao, T. (2017, October 17). Heat flow and zeroes of polynomials. What's New. https: //terrytao.wordpress.com/2017/10/17/heat-flow-and-zeroes-of-polynomials/

[2]. Tang, T. (1993). The Hermite spectral method for Gaussian-type functions.SIAM Journal on Scientific Computing, 14(3), 594–606.

[3]. Berg, C. (2008). The matrix moment problem. In A. J. P. L. Branquinho & A. P. Foulquié Moreno (Eds.), Coimbra lecture notes on orthogonal polynomials (pp. 1–58). Nova Science Publishers.

[4]. Damanik, D., Pushnitski, A., & Simon, B. (2008). The analytic theory of matrix orthogonal polynomials.Surveys in Approximation Theory, 4, 1–85.

[5]. Hahn, W. (1935). Über die Jacobischen Polynome und zwei verwandte Polynomklassen.Mathematische Zeitschrift, 39(1), 634–638. https: //doi.org/10.1007/BF01201380

References

[1]. Tao, T. (2017, October 17). Heat flow and zeroes of polynomials. What's New. https: //terrytao.wordpress.com/2017/10/17/heat-flow-and-zeroes-of-polynomials/

[2]. Tang, T. (1993). The Hermite spectral method for Gaussian-type functions.SIAM Journal on Scientific Computing, 14(3), 594–606.

[3]. Berg, C. (2008). The matrix moment problem. In A. J. P. L. Branquinho & A. P. Foulquié Moreno (Eds.), Coimbra lecture notes on orthogonal polynomials (pp. 1–58). Nova Science Publishers.

[4]. Damanik, D., Pushnitski, A., & Simon, B. (2008). The analytic theory of matrix orthogonal polynomials.Surveys in Approximation Theory, 4, 1–85.

[5]. Hahn, W. (1935). Über die Jacobischen Polynome und zwei verwandte Polynomklassen.Mathematische Zeitschrift, 39(1), 634–638. https: //doi.org/10.1007/BF01201380

Cite this article

Liu,J. (2025). An analytic way to prove the explicit formula for Hermite polynomial after heat flow deformation and observation in 3D dimensions. Advances in Operation Research and Production Management,4(3),35-52.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Journal: Advances in Operation Research and Production Management

Volume number: Vol.4

Issue number: Issue 3

ISSN: 3029-0880(Print) / 3029-0899(Online)

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