# Some Miscellaneous Results on Hermite Polynomials

Julius Lehmann

► To cite this version:

Julius Lehmann. Some Miscellaneous Results on Hermite Polynomials. 2026. hal-05488698

HAL Id: hal-05488698

https://hal.science/hal-05488698v1

Preprint submitted on 2 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Some Miscellaneous Results on Hermite Polynomials

Julius Lehmann*

Physics of Complex Biosystems, Technical University of Munich, 85748 Garching,
Germany

February 2, 2026

## Abstract

We derive a general integral identity relating Hermite and Gegenbauer polynomials. This identity yields explicit Laplace and Mellin transform pairs and leads to closed-form Hermite series expansions for classes of functions, including power functions and the Heaviside step function. Using the Ramanujan master theorem, we obtain an explicit power-series representation of Gaussian-weighted Hermite polynomials and apply the result to derive the series expansion of the eigenfunctions of the quantum harmonic oscillator.

**Keywords:** Hermite polynomials; Gegenbauer polynomials; Hermite expansion; Quantum harmonic oscillator

## 1 Main Identity

The Hermite polynomials $H_n(x)$ are a classical family of orthogonal polynomials that were first introduced by Laplace and later studied in detail by Chebyshev [10, 1]. Subsequently, Hermite investigated these polynomials further and popularized them, leading to their modern nomenclature. They are most conveniently defined through their generating function [1, 3, 10]

$$e^{-t^2+2xt} = \sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n \quad (1)$$

together with the useful identity

$$H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}. \quad (2)$$

Hermite polynomials possess numerous applications arising, for example, in signal processing, probability theory, and as the eigenstates of the quantum harmonic oscillator, among many others. Another classical family of orthogonal polynomials are the Gegenbauer polynomials of order $\alpha$, denoted by $C_n^{(\alpha)}(x)$, which generalize the Legendre polynomials and

<footnote>*e-mail: julius.lehmann(at)tum.de</footnote>
are defined through the generating function [10, 1, 3]

$$ \frac{1}{(t^2 - 2xt + 1)^\alpha} = \sum_{n=0}^{\infty} C_n^{(\alpha)}(x) t^n. \qquad (3) $$

These two families of polynomials are related by a limit relation [8]

$$ \lim_{\alpha \to \infty} \alpha^{-\frac{n}{2}} C_n^{(\alpha)} \left( \frac{x}{\sqrt{\alpha}} \right) = \frac{H_n(x)}{n!}, \qquad (4) $$

and can be expressed in terms of one another [5]. Using these definitions, we derive a rather general integral that proves useful in constructing Hermite expansions and related series. Starting from the generating function of the Gegenbauer polynomials and employing the Schwinger parameterization, we obtain

$$ \frac{1}{(t^2 - 2st + 1)^m} = \frac{1}{\Gamma(m)} \int_0^\infty dx \, x^{m-1} e^{-x(t^2 - 2st + 1)}. \qquad (5) $$

The integrand can be expanded in terms of Hermite polynomials, which, after an appropriate change of variables, yields

$$ \frac{1}{(t^2 - 2st + 1)^m} = \sum_{n=0}^{\infty} t^n \left( \frac{2s^{-2m-n}}{n!\Gamma(m)} \int_0^\infty dx \, e^{-\frac{x^2}{s^2}} x^{2m+n-1} H_n(x) \right). \qquad (6) $$

By comparing coefficients and setting $2m+n-1=k$, we arrive at the final expression for the integral

$$ \int_0^\infty dx \, e^{-\frac{x^2}{s^2}} x^k H_n(x) = \frac{n!}{2} \Gamma\left(\frac{k-n+1}{2}\right) C_n^{\left(\frac{k-n+1}{2}\right)}(s) s^{k+1}. \qquad (7) $$

This integral establishes a direct connection between the Hermite and Gegenbauer polynomials. A notable special case arises by setting $k=n+1$, which leads to

$$ \int_0^\infty dx \, e^{-\frac{x^2}{s^2}} x^{n+1} H_n(x) = \frac{n!}{2} U_n(s) s^{n+2}, \qquad (8) $$

where $U_n(x)$ denote the Chebyshev polynomials of the second kind [6, 1].

## 2 Applications

Several straightforward applications follow directly from this integral. First, after a suitable change of variables, we observe that it implies the Laplace transform pair

$$ \mathcal{L}\{x^k H_n(\sqrt{x})\}(s) = n! \Gamma\left(k - \frac{n}{2} + 1\right) C_n^{\left(k - \frac{n}{2} + 1\right)}\left(\frac{1}{\sqrt{s}}\right) \frac{1}{s^{k+1}}. \qquad (9) $$
On the other hand, the Hermite polynomials are orthogonal with respect to the weight $e^{-x^2}$, which implies the relation [10]

$$ \int_{-\infty}^{\infty} dx \, e^{-x^2} H_m(x) H_n(x) = \sqrt{\pi n! 2^n} \delta_{m,n}. \qquad (10) $$

As a consequence, any suitable function $f(x)$ can be expanded in terms of Hermite polynomials as [7]

$$ f(x) = \sum_{n=0}^{\infty} c_n H_n(x), \qquad (11) $$

where the expansion coefficients are given by

$$ c_n = \frac{1}{\sqrt{\pi n! 2^n}} \int_{-\infty}^{\infty} dx \, e^{-x^2} H_n(x) f(x). \qquad (12) $$

Rearranging the integrand, we obtain

$$ \int_{0}^{\infty} dx \, e^{-x^2} H_n(x) e^{-\left(\frac{1}{s^2}-1\right)x^2} x^k. \qquad (13) $$

After redefining $\frac{1}{s^2}-1 = \alpha$, this expression is proportional to the coefficients in the Hermite series expansion of

$$ f(x) = \begin{cases} e^{-\alpha x^2} x^k & x \ge 0 \\ 0 & x < 0. \end{cases} \qquad (14) $$

This observation implies that, in the region of convergence, the Hermite series of $f(x)$ takes the form

$$ \frac{1}{\sqrt{\pi}} \sum_{n=0}^{\infty} \frac{\Gamma(\frac{k-n+1}{2})}{2^{n+1}} \frac{C_n^{(\frac{k-n+1}{2})} (\frac{1}{\sqrt{1+\alpha}})}{(1+\alpha)^{\frac{k+1}{2}}} H_n(x) = \begin{cases} e^{-\alpha x^2} x^k & x \ge 0 \\ 0 & x < 0, \end{cases} \qquad (15) $$

where in figure 1 we demonstrate an example of this series. Several special cases follow immediately, for instance, setting $\alpha = 0$ yields

$$ \frac{1}{\sqrt{\pi}} \sum_{n=0}^{\infty} \binom{k}{n} \frac{\Gamma(\frac{k-n+1}{2})}{2^{n+1}} H_n(x) = \begin{cases} x^k & x \ge 0 \\ 0 & x < 0. \end{cases} \qquad (16) $$

Furthermore, taking the limit $k \to 0$ in this expression produces the Hermite expansion of the step function,

$$ \frac{1}{2} \sum_{n=0}^{\infty} \frac{H_n(x)}{n! \Gamma(1 - \frac{n}{2})} = \begin{cases} 1 & x > 0 \\ \frac{1}{2} & x = 0 \\ 0 & x < 0. \end{cases} \qquad (17) $$
<figure><img src="image_2.png" /><figcaption>Figure 1: Hermite polynomial expansion of $e^{-\alpha x^2}x^k$ for $x \ge 0$ and zero otherwise with $\alpha = \frac{1}{4}$ and $k = \frac{5}{2}$ (black dashed line) for partial sums 2, 5, 10, 30.</figcaption></figure>

We can further simplify this result by noting that [11]

$$ \frac{1}{\Gamma(1 - \frac{n}{2})} = \begin{cases} 0 & n \text{ even} \\ \frac{(2n)!}{(-4)^{n}n!\sqrt{\pi}} & n \text{ odd,} \end{cases} \qquad (18) $$

which allows us to rewrite the series as

$$ \frac{1}{2} + \frac{1}{\sqrt{\pi}} \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n + 1)n!2^{2n+1}} H_{2n+1}(x) = \begin{cases} 1 & x > 0 \\ \frac{1}{2} & x = 0 \\ 0 & x < 0. \end{cases} \qquad (19) $$

This expression was also obtained, albeit with a minor error in the numerator, using a different approach in [7].

Another immediate consequence of the integral (7) is the Mellin transform pair

$$ \mathcal{M}\left\{e^{-\frac{x^2}{s^2}} H_n(x)\right\}(k) = \frac{n!}{2} \Gamma\left(\frac{k-n}{2}\right) C_n^{\left(\frac{k-n}{2}\right)}(s) s^k, \qquad (20) $$

where the Mellin transform is defined by [4]

$$ \mathcal{M}\{f(x)\}(k) = \int_{0}^{\infty} dx \, x^{k-1} f(x). \qquad (21) $$

According to the Ramanujan master theorem [9, 2], if a function can be expressed as

$$ f(x) = \sum_{n=0}^{\infty} \frac{\phi(n)}{n!} (-x)^n, \qquad (22) $$
then its Mellin transform satisfies the simple relation

$$ \int_0^\infty dx \, x^{k-1} f(x) = \Gamma(k) \phi(-k), \qquad (23) $$

where $\phi(-k)$ is understood as the analytic continuation of the discrete sequence $\phi(n)$ to complex arguments. Applying this result in reverse, we can formally relate the expansion coefficients in

$$ e^{-\frac{x^2}{s^2}} H_n(x) = \sum_{n=0}^{\infty} \frac{\phi(n)}{n!} (-x)^n \qquad (24) $$

to the right-hand side of Eq. (20), obtaining

$$ \phi(k) = \frac{n! \Gamma(-\frac{k+n}{2})}{2 \Gamma(-k) s^k} C_n(-\frac{k+n}{2})(s). \qquad (25) $$

The ratio $\frac{\Gamma(-\frac{k+n}{2})}{\Gamma(-k)}$ is as such undefined at integer values of $k$ and must therefore be interpreted by taking the limit. Accordingly, using the standard result [11]

$$ \Gamma(-n + \varepsilon) = \frac{(-1)^n}{n!} \frac{1}{\varepsilon} + \dots, \qquad (26) $$

we find for the limit

$$ \lim_{k \to m \in \mathbb{Z}} \frac{\Gamma(-\frac{k+n}{2})}{\Gamma(-k)} = \lim_{\varepsilon \to 0} \frac{\Gamma(-\frac{m+n}{2} + \frac{\varepsilon}{2})}{\Gamma(-m + \varepsilon)} = \begin{cases} (-1)^{\frac{m+n}{2} - m} \frac{m!}{\binom{m+n}{2}!} & m+n \text{ even} \\ 0 & m+n \text{ odd,} \end{cases} \qquad (27) $$

where we introduced $k = m - \varepsilon$. Consequently, the series expansion can be written as

$$ e^{-\frac{x^2}{s^2}} H_n(x) = \frac{n!}{2} \sum_{\substack{k+n \\ \text{even}}} (-1)^{\frac{n-k}{2}} \frac{C_n\left(-\frac{k+n}{2}\right)(s)}{s^k \binom{k+n}{2}!} (-x)^k. \qquad (28) $$

This function is proportional to the so-called Hermite functions and, for an appropriate choice of $s$, is proportional to the eigenfunctions of the quantum harmonic oscillator. In particular, for a quantum harmonic oscillator of frequency $\omega$ and mass $m$, the eigenfunctions are given by [12]

$$ \psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}} e^{-\frac{m\omega}{2\hbar}x^2} H_n\left(\sqrt{\frac{m\omega}{\hbar}}x\right), \qquad (29) $$

which immediately implies $s = \sqrt{2}$ and yields the series expansion of $\psi_n(x)$ about $x = 0$ in the form

$$ \psi_n(x) = \frac{1}{2} \sqrt{\frac{n!}{2^n}} \left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}} \sum_{\substack{k+n \\ \text{even}}} (-1)^{\frac{n-k}{2}} \frac{C_n\left(-\frac{k+n}{2}\right)(\sqrt{2})}{2^{\frac{k}{2}} \binom{k+n}{2}!} \left(-\sqrt{\frac{m\omega}{\hbar}}x\right)^k. \qquad (30) $$
References

[1] Milton Abramowitz and Irene A Stegun. *Handbook of mathematical functions with formulas, graphs, and mathematical tables*. Vol. 55. US Government printing office, 1948.

[2] Tewodros Amdeberhan et al. "Ramanujan's master theorem". In: *The Ramanujan Journal* 29.1 (2012), pp. 103–120.

[3] Harry Bateman and Bateman Manuscript Project. *Higher Transcendental Functions [Volumes I-III]*. McGraw-Hill Book Company, 1953.

[4] Jacqueline Bertrand, Pierre Bertrand, and Jean-Philippe Ovarlez. "The mellin transform". In: *The transforms and applications handbook* (1995).

[5] Juan Carlos López Carreno and Rosalba Mendoza Suárez. "Connection between the Hermite and Gegenbauer polynomials using the Lewanowicz method". In: *Computer Science* 18.4 (2023), pp. 627–636.

[6] Pafnuty Chebyshev. *Théorie des mécanismes connus sous le nom de parallélogrammes*. fr. Imprimerie de L'Académie Impériale des Sciences, 1853. DOI: 10.3931/E-RARA-120037. URL: https://www.e-rara.ch/doi/10.3931/e-rara-120037.

[7] Tom P Davis. "A general expression for Hermite expansions with applications". In: *The Mathematics Enthusiast* 21.1 (2024), pp. 71–87.

[8] F. W. J. Olver et al. *NIST Digital Library of Mathematical Functions*. http://dlmf.nist.gov/, Release 1.2.5 of 2025-12-15. 2020. URL: https://dlmf.nist.gov.

[9] Katarzyna Górska et al. "The Ramanujan master theorem and its implications for special functions". In: *Applied Mathematics and Computation* 218.23 (2012), pp. 11466–11471.

[10] Gábor Szegő. *Orthogonal Polynomials*. Vol. 23. American Mathematical Society, 1939.

[11] E. Weisstein. *Gamma Function*. From *MathWorld—A Wolfram Web Resource*. Last visited on 29/1/2026. URL: https://mathworld.wolfram.com/GammaFunction.html.

[12] Barton Zwiebach. *Mastering quantum mechanics: essentials, theory, and applications*. MIT press, 2022.