Old and New Results About Relativistic Hermite Polynomials

TL;DR

This paper reviews and extends the theory of relativistic Hermite polynomials by providing new proofs and results, utilizing moments, subordination, and Nagel's identity, connecting them to Gegenbauer polynomials.

Contribution

It offers new proofs and novel results about relativistic Hermite polynomials, enhancing understanding of their properties and relations to classical polynomials.

Findings

New proofs of known properties of RHP

New results about RHP properties

Connections between RHP and Gegenbauer polynomials

Abstract

The relativistic Hermite polynomials (RHP) were introduced in 1991 by Aldaya et al. in a generalization of the theory of the quantum harmonic oscillator to the relativistic context. These polynomials were later related to the more classical Gegenbauer (or ultraspherical) polynomials in a study by Nagel. Thus some of their properties can be deduced from the properties of the well-known Gegenbauer polynomials, as underlined by M. Ismail. In this report we give new proofs of already known results but also new results about these polynomials. We use essentially three basic tools: the representation of polynomials as moments, the subordination tool and Nagel's identity.

Equations

$$H_n^N\left(X\right)={\left(-1\right)}^n{\left(1+\frac{X^2}{N}\right)}^{N+n}\frac{d^n}{d{X}^n}{\left(1+\frac{X^2}{N}\right)}^{-N};$$

$$H_0^N\left(X\right)=1;H_1^N\left(X\right)=2X;H_2^N\left(X\right)=2\left(-1+X^2\left(2+\frac{1}{N}\right)\right)$$

$$H_3^N\left(X\right)=4\left(1+\frac{1}{N}\right)\left(X^3\left(2+\frac{1}{N}\right)-3X\right)$$

$$H_n\left(X\right)={\left(-1\right)}^n\exp \left(X^2\right)\frac{d^n}{d{X}^n}\exp \left(-X^2\right)$$

$$\underset{N\to +\infty }{\lim }{H}_n^N\left(X\right)=H_n\left(X\right).$$

$$H_n^N\left(X\right)=\frac{{\left(2N\right)}_n}{{\left(2\sqrt{N}\right)}^n}\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }\frac{{\left(-1\right)}^k}{{\left(N+\frac{1}{2}\right)}_k}\frac{n!}{\left(n-2k\right)!k!}{\left(\frac{2X}{\sqrt{N}}\right)}^{n-2k}$$

$$C_n^N\left(X\right)={\gamma }_n^N{\left(-1\right)}^n{\left(1-X^2\right)}^{\frac{1}{2}-N}\frac{d^n}{d{X}^n}{\left(1-X^2\right)}^{n+N-\frac{1}{2}}$$

$${\gamma }_n^N=\frac{{\left(2N\right)}_n}{2^{n}n!{\left(N+\frac{1}{2}\right)}_n};$$

$$C_0^N\left(X\right)=1;C\left(X\right)=2NX;C\left(X\right)=2N\left(N+1\right)X^2-N;C_3^N\left(X\right)=2N\left(N+1\right)\left(\frac{2\left(N+2\right)}{3}{X}^3-X\right)$$

$$C_n^N\left(X\right)=\sum _{k=0}^{\lfloor \frac{n}{2}\rfloor }{\left(-1\right)}^k\frac{{\left(N\right)}_{n-k}}{\left(n-2k\right)!k!}{\left(2X\right)}^{n-2k}.$$

$$H_n^N\left(X\sqrt{N}\right)=\frac{n!}{N^{\frac{n}{2}}}{\left(1+X^2\right)}^{\frac{n}{2}}{C}_n^N\left(\frac{X}{\sqrt{1+X^2}}\right).$$

$$C_n^N\left(X\right)={\alpha }_n^N{H}_n^{\frac{1}{2}-N-n}\left(-iX\sqrt{\frac{1}{2}-N-n}\right)$$

$${\alpha }_n^N={\left(-2i\right)}^n\frac{{\left(N\right)}_n}{{\left(2N+n\right)}_n}\frac{{\left(\frac{1}{2}-N-n\right)}^{\frac{n}{2}}}{n!}.$$

$$C_n^N\left(iX\right)={\left(-2i\right)}^n\frac{{\left(N\right)}_n}{{\left(2N+n\right)}_n}{\left(1+X^2\right)}^{\frac{n}{2}}{C}_n^{\frac{1}{2}-N-n}\left(\frac{X}{\sqrt{1+X^2}}\right)$$

$$E_b{H}_n\left(X\sqrt{b}\right)={\int }_0^{+\infty }{H}_n\left(X\sqrt{b}\right)f\left(b\right)db.$$

$$H_n\left(X\right)=\frac{N^{\frac{n}{2}}}{{\left(N\right)}_{\frac{n}{2}}}{E}_c{H}_n^N\left(\frac{X\sqrt{N}}{\sqrt{c}}\right),c\sim \Gamma \left(N+\frac{n+1}{2}\right).$$

$$C_n^N\left(X\right)=\frac{{\left(N\right)}_{\frac{n}{2}}}{n!}{E}_b{H}_n\left(X\sqrt{b}\right),b\sim \Gamma \left(N+\frac{n}{2}\right)$$

$$E{c}^l={\left(N+\frac{n+1}{2}\right)}_l$$

$$E_c{H}_n^N\left(\frac{X\sqrt{N}}{\sqrt{c}}\right)$$

$$\frac{{\left(N\right)}_{\frac{n}{2}}}{n!}{E}_b{H}_n\left(X\sqrt{b}\right)$$

$$H_n\left(X\right)=2^n{E}_Z{\left(X+iZ\right)}^n,$$

$$C_n^N\left(X\right)=\frac{{\left(2N\right)}_n}{n!}{E}_{Z_N}{\left[X+i\sqrt{1-X^2}{Z}_N\right]}^n$$

$$f_{Z_N}\left(Z\right)=\frac{\Gamma \left(N+\frac{1}{2}\right)}{\Gamma \left(N\right)\Gamma \left(\frac{1}{2}\right)}{\left(1-Z^2\right)}^{N-1},-1\le Z\le +1.$$

$$H_n^N\left(X\right)=\frac{{\left(2N\right)}_n}{N^{\frac{n}{2}}}{E}_{Z_N}{\left(\frac{X}{\sqrt{N}}+i{Z}_N\right)}^n$$

$$C_n^N\left(X\right)=\frac{2^n{\left(N\right)}_{\frac{n}{2}}}{n!}{E}_{Z,b}{\left(X\sqrt{b}+iZ\right)}^n$$

$$H_n^N\left(X\sqrt{N}\right)=\frac{2^n{\left(2N\right)}_n}{N^{\frac{n}{2}}}{E}_{b,Z}{\left(X\sqrt{b}+i\sqrt{1+X^2}Z\right)}^n$$

$$C_n^N\left(X\right)=\frac{1}{n!}E{\left[\left(X+\sqrt{X^2-1}\right)U+\left(X-\sqrt{X^2-1}\right)V\right]}^n$$

$$C_n^N\left(X\right)=\frac{E{\left(U+V\right)}^n}{n!}{E}_{Z_N}{\left[X+i\sqrt{1-X^2}{Z}_N\right]}^n=\frac{1}{n!}E{\left[X\left(U+V\right)+i\sqrt{1-X^2}{Z}_N\left(U+V\right)\right]}^n$$

$$Z_N=\frac{U-V}{U+V}$$

$$C_n^N\left(X\right)=\frac{1}{n!}E{\left[X\left(U+V\right)+i\sqrt{1-X^2}\left(U-V\right)\right]}^n$$

$$H_n^N\left(X\sqrt{N}\right)=\frac{1}{N^{\frac{n}{2}}}E{\left[\left(i+X\right)U+\left(-i+X\right)V\right]}^n$$

Full text

Old and new results about relativistic Hermite polynomials

C. Vignat

1. Introduction

The relativistic Hermite polynomials (RHP) were introduced in 1991 by Aldaya et al. [1] in a generalization of the theory of the quantum harmonic oscillator to the relativistic context. These polynomials were later related to the more classical Gegenbauer (or ultraspherical) polynomials in a study by Nagel [2]. Thus some of their properties can be deduced from the properties of the well-known Gegenbauer polynomials, as underlined by M. Ismail in [3]. In this report we give new proofs of already known results but also new results about these polynomials. We use essentially three basic tools: the representation of polynomials as moments, the subordination tool and Nagel’s identity.

2. Definitions and Tools

2.1. Polynomials

2.1.1. Relativistic Hermite polynomials

The relativistic Hermite polynomial of degree $n$ and parameter $N\neq 0$ is defined by the Rodrigues formula

[TABLE]

examples of RHP polynomials are

[TABLE]

These polynomials are extensions of the classical Hermite polynomials $H_{n}\left(X\right)$ that are defined by the Rodrigues formula

[TABLE]

and thus can be obtained as the limit case

[TABLE]

An explicit formula for the relativistic Hermite polynomial is [1]

[TABLE]

where $\left(n\right)_{k}$ is the Pochhammer symbol.

2.1.2. Gegenbauer polynomials

The Gegenbauer polynomial of degree $n$ and parameter $N$ is defined by the Rodrigues formula

[TABLE]

with

[TABLE]

examples of Gegenbauer polynomials are

[TABLE]

and an explicit formula is

[TABLE]

2.2. tools

2.2.1. Nagel’s identity

Nagel’s identity [2] is a first link between relativistic Hermite polynomials and Gegenbauer polynomials. The RHP with degree $n$ and parameter $N$ is linked to the Gegenbauer polynomial with same degree and same parameter via Nagel’s identity

[TABLE]

As a consequence of Nagel’s identity, we deduce the following theorem

Theorem 1**.**

The relativistic Hermite polynomials is related to the Gegenbauer polynomial as

[TABLE]

with

[TABLE]

Proof.

By the identity [7, (7.2.15.8)],

[TABLE]

and the result follows by application of Nagel’s identity. ∎

Formula (2.3) was derived by Nagel [2], who notes: “This representation does not seem to be very useful however.” Indeed, the fact that the parameter $N$ of the Gegenbauer polynomial is transformed, for the Relativistic Hermite polynomial, into a parameter $\frac{1}{2}-N-n$ that depends on the degree $n$, seems to make this identity a priori less useful than expected. However, as will be shown in Section 5, this formula allows in fact an easy extension of the scaling identity for Gegenbauer polynomials to the case of Relativistic Hermite polynomials.

A similar version of formula (2.3) appears in [8] and also in [9] in the case of deformed Hermite polynomials.

2.2.2. The subordination tool

Writing polynomials as scale mixtures of others allows to deduce properties between them. We use the probabilistic expectation to denote the subordination by the mesure $f$ as

[TABLE]

The subordination dependence between Hermite, relativistic Hermite and Gegenbauer polynomials is as follows.

Theorem 2**.**

The Hermite, relativistic Hermite and Gegenbauer polynomials are related as follows

[TABLE]

[TABLE]

Here, $b\sim\Gamma\left(N+\frac{n}{2}\right)$ means that $b$ is a random variable that follows the Gamma distribution with shape parameter $N+\frac{n}{2}$ that is $\gamma_{N+\frac{n}{2}}\left(b\right)=\frac{1}{\Gamma\left(N+\frac{n}{2}\right)}e^{-b}b^{N+\frac{n}{2}-1},\,\,b\geq 0$.

Proof.

Since

[TABLE]

we deduce from (2.1) that

[TABLE]

so that the first result follows after application of Euler’s duplication formula.

The same way, we compute

[TABLE]

which coincides with (2.2). ∎

2.2.3. The moment representation

The well-known moment representation of the Hermite polynomial

[TABLE]

where $Z$ is Gaussian centered with variance $\frac{1}{2}$ is a consequence of the integral formula [5, 8.951].

Its extension to the Gegenbauer polynomials is deduced from integral [5, 8.931] (called Laplace first integral in [3]) and reads

[TABLE]

where the random variable $Z_{N}$ has a Student-r distribution

[TABLE]

Using Nagel’s identity, we deduce from (2.6) the moment representation for RHP polynomials

[TABLE]

with $Z_{N}$ distributed according to (2.7).

Another moment representation for the Gegenbauer polynomials is as follows:

Theorem 3**.**

The Gegenbauer polynomial and relativistic Hermite polynomials have moment representation

[TABLE]

and

[TABLE]

where $Z$ is Gaussian centered with variance $\frac{1}{2}$ and independent of $b$ which is Gamma distributed with shape parameter $N+\frac{n}{2}.$

Proof.

This expression is derived by the application of the subordination identity (2.4) to the moment representation (2.5). The representation (2.10) is deduced from (2.9) using Nagel’s identity. ∎

However, another set of moment representations involving two random variables can be deduced from (2.6) as follows:

Theorem 4**.**

The Gegenbauer polynomial has for moment representation

[TABLE]

where $U$ and $V$ are independently distributed according to a Gamma law with shape parameter $N.$

Proof.

Consider $U$ and $V$ independently distributed according to a Gamma law with shape parameter $N;$ then $U+V$ is Gamma distributed with shape parameter $2N$ so that $E\left(U+V\right)^{n}=\left(2N\right)_{n}$. We deduce from (2.6) that

[TABLE]

but a well-known stochatic representation for $Z_{N}$ is

[TABLE]

where $Z_{N}$ is independent of $\left(U+V\right)$ so that

[TABLE]

and the result follows. ∎

From this result we deduce

Theorem 5**.**

A moment representation for the relativistic Hermite polynomial is

[TABLE]

where $U$ and V are independently distributed according to a Gamma law with shape parameter $N.$

Proof.

This is a direct consequence of Nagel’s formula.∎

Remark 6*.*

Representation (2.8) can proved directly from the explicit expression (2.1) of the Relativistic Hermite polynomials: since $Z_{N}$ has odd moments equal to zero, we have

[TABLE]

the even moments can be computed as

[TABLE]

so that

[TABLE]

which coincides with (2.1).

We note moreover that the moment representation (2.11) was derived recently by Sun [4] using another proof based on the generating function of the Gegenbauer polynomials.

2.2.4. example of application

As an example of the usefulness of the moment representations given above, we derive the famous [5, 8.952.1]

[TABLE]

and its relativistic version

[TABLE]

In the Gegenbauer case, we rather use the stochastic representation (2.4) to obtain

[TABLE]

and since $E_{b\sim\Gamma_{N+\frac{n}{2}}}\sqrt{b}f\left(b\right)=\frac{\Gamma\left(N+\frac{n}{2}+\frac{1}{2}\right)}{\Gamma\left(N+\frac{n}{2}\right)}E_{c\sim\Gamma_{N+\frac{n+1}{2}}}f\left(c\right),$ we deduce

[TABLE]

which coincides with [5, 8.935.2].

3. The Gram-Schmidt operator

A family of orthogonal polynomials can be obtained by applying the Gram-Schmidt operator to the canonical basis $\left\{1,X,\dots,X^{n}\right\}.$ We show here how this operator can be expressed in the case where a moment formula exists.

Theorem 7**.**

If a polynomial $P_{n}\left(X\right)$ can be expressed as

[TABLE]

for some random variable $Z$ then

[TABLE]

where $\phi_{Z}\left(u\right)=E_{Z}\exp\left(iuZ\right)$ is the characteristic function of $Z.$

Proof.

By definition

[TABLE]

∎

As an application of this theorem, we recover the following well-known result for the Hermite polynomials

[TABLE]

The extension of this result to the case of the Relativistic Hermite polynomials is as follows

Theorem 8**.**

The Gram-Schmidt operator associated to the relativistic Hermite polynomial is

[TABLE]

where the normalized Bessel function is

[TABLE]

Proof.

The characteristic function of the random variable $Z_{N}$ in (2.8) is the normalized Bessel function

[TABLE]

∎

As $N\to+\infty,$ since

[TABLE]

we recover the classical Hermite case.

4. addition theorems

4.1. A new proof of the classical addition theorem for Hermite polynomials

The summation theorem for Hermite polynomials [5, 8.958.1] states that

[TABLE]

We give here a short proof using the moment representation; we assume first that $\sum_{k=1}^{r}a_{k}^{2}=1$ so that we expand

[TABLE]

where variables $Z_{k}$ are independent and Gaussian with variance $\frac{1}{2}$ so that $\sum_{k=1}^{r}a_{k}Z_{k}$ is Gaussian with variance $\frac{1}{2}$ and we deduce

[TABLE]

so that the result follows. Now we replace $a_{k}$ by $\frac{a_{k}}{\sqrt{\sum_{l=1}^{r}a_{l}^{2}}}$ in this equality so that we obtain the general case (4.1).

4.2. An addition theorem for the relativistic Hermite polynomial

From the well-known addition formula [7, 7.2.13.36] for Gegenbauer polynomials

[TABLE]

we deduce the following

Theorem 9**.**

An addition theorem for the relaivistic Hermite polynomials is

[TABLE]

Proof.

Using formula (2.3), we deduce

[TABLE]

Replacing $X$ by $iX$ and $Y$ by $iX$ and computing

[TABLE]

so that, replacing $X$ by $iX$ and $Y$ by $iY$ yields

[TABLE]

and the result is obtained by replacing $\frac{1}{2}-N-n$ by $N.$ ∎

5. The scaling identity

The scaling identity for Hermite polynomials reads [3, 4.6.33]

[TABLE]

A quick proof can be given using the moment representation (2.5):

[TABLE]

where $Z_{1}$ and $Z_{2}$ are independent Gaussian random variables with variance $\frac{1}{2}$ so that

[TABLE]

since the odd moments of a Gaussian are null; as moreover $EZ_{2}^{2l}=2^{2l-1}\frac{\Gamma\left(2l\right)}{\Gamma\left(l\right)}$ we deduce the result.

It is possible to extend this proof to the case of Gegenbauer polynomials using either the moment representation (2.6) or (2.9); however, a more simple proof can be derived using the subordination relation (2.4) as follows.

Theorem 10**.**

[7*, 7.2.13.37]*The scaling identity for Gegenbauer polynomials reads

[TABLE]

Proof.

From the subordination formula (2.4) and the scaling formula (5.1) we obtain (where the notation $b_{N+\frac{n}{2}}$ is a shortcut for $b\sim\Gamma_{N+\frac{n}{2}}$)

[TABLE]

but

[TABLE]

with $n^{\prime}=n-2l$ and $N^{\prime}=N+l$ so that

[TABLE]

∎

The scaling identity for the RHP can be deduced from the preceding one using formula (2.3).

Theorem 11**.**

The scaling identity for relativistic Hermite polynomials is

[TABLE]

Proof.

Starting from (5.2) and using (2.3), we deduce

[TABLE]

A short computation yields to

[TABLE]

Replacing $N$ by $\frac{1}{2}-n-N$ we deduce

[TABLE]

and the result follows from the fact that

[TABLE]

∎

6. Generating Functions

6.1. the generating function for the RHP

The generating function for the RHP is computed in [6] using a differential equation; we note that it can not be obtained directly using formula (2.3). However, it can be easily obtained from the moment representation (2.13) or (2.8) as follows.

Theorem 12**.**

the generating function for the RHP reads

[TABLE]

for $|t|<\frac{\sqrt{N}}{\sqrt{1+\frac{X^{2}}{N}}}.$

Proof.

Starting from (2.13) we obtain

[TABLE]

with $E_{U}\exp\left(\lambda U\right)=\left(1-\lambda\right)^{-N}$ for $|\lambda|<1$ so that, for $|t|<\frac{\sqrt{N}}{\sqrt{1+\frac{X^{2}}{N}}},$

[TABLE]

We remark that we recover the generating funtion for Hermite polynomials as $N\to+\infty.$ The proof using the moment representation (2.8) reads

[TABLE]

for $|t|<\frac{\sqrt{N}}{\sqrt{1+\frac{X^{2}}{N}}}.$ But from [5, 3.665.1]

[TABLE]

so that the result follows with $a=1-\frac{tX}{N}$ and $b=\frac{t}{\sqrt{N}}.$ ∎

We note that formula [5, 3.665.1] can be reovered using only probabilistic tools as follows: since $\left(1-u\right)^{-2N}=E_{W_{2N}}\exp\left(uW\right)$ is the moment generating function of a Gamma random variable $W_{2N}$ with shape parameter $2N,$ we deduce

[TABLE]

But, by (2.12),

[TABLE]

where $U$ and $V$ are two independent Gamma random variables with shape parameter equal to $N.$ Thus

[TABLE]

and the result follows.

6.2. Feldheim and Villenkin

We give here a short proof of the Feldheim Villenkin generating function for the normalized Gegenbauer polynomials

[TABLE]

where the function $j_{N-\frac{1}{2}}$ is defined as in (3.1), by remarking that, using the moment representation (2.6),

[TABLE]

so that

[TABLE]

The latest expectation is nothing but the characteristic function $\phi_{Z_{N}}\left(u\right)=j_{N-\frac{1}{2}}\left(u\right)$ of $Z_{N}$ computed at $u=r\sin\theta,$ so that the result follows.

An equivalent result for the RHP is as follows:

Theorem 13**.**

A generating function for the relativistic Hermite polynomials is

[TABLE]

Proof.

The proof follows the same line as the one above, starting from the moment representation (2.8). ∎

6.3. Another generating function for the Hermite polynomials

A classical generating function for the Hermite polynomials [3, 4.6.29] reads

[TABLE]

where $\phi\left(X,t\right)=\exp\left(2Xt-t^{2}\right)$ is the generating function of the Hermite polynomials.

A generalization of this formula to the relativistic Hermite polynomials reads as follows.

Theorem 14**.**

For the relativistic Hermite polynomials,

[TABLE]

where $\phi_{N}\left(X,t\right)=\left(1-\frac{2Xt}{N}+\frac{X^{2}t^{2}}{N^{2}}+\frac{t^{2}}{N^{2}}\right)^{-N}$ is the generating function of the relativistic Hermite polynomials.

Proof.

Denote

[TABLE]

so that

[TABLE]

and, using the moment representation (2.13),

[TABLE]

Define $\beta=\sqrt{1+\frac{X^{2}}{N}}$ and $z=\beta^{2}\left(t-\frac{X}{\beta^{2}}\right)$ so that $\phi_{N}\left(X,t\right)=\beta^{2N}\left(1+\frac{z^{2}}{N}\right)^{-N}$ and

[TABLE]

so that the result holds. ∎

7. Determinants

Determinants with orthogonal polynomials entries have been extensively studied [11] by Karlin and Szegö and recently revisited by Ismail [12]. We show here that, in the case of Turàn determinants, the moment representation derived above allows to extend some of these results to relativistic Hermite polynomials.

We propose a method slightly different from the one used in [12] based on the following result.

Theorem 15**.**

If the polynomials $P_{n}\left(X\right)$ can be expressed as

[TABLE]

for some random variable $Z$ then the Turàn determinant

[TABLE]

is a constant equal to

[TABLE]

Proof.

We use the formula of Wilks [13]

[TABLE]

where $m_{k}=EU_{0}^{k}$ and the $U_{i}$ are independent and identically distributed. Thus since

[TABLE]

with $U=X+iZ,$ we deduce

[TABLE]

so that

[TABLE]

and the result follows. ∎

Using this result, we deduce the Turàn determinant for the normalized Relativistic Hermite polynomial defined as

[TABLE]

Theorem 16**.**

The Turàn determinant for the normalized polynomial (7.1) is a constant equal to

[TABLE]

Proof.

From Theorem 15, we have

[TABLE]

where, from (2.8), each $Z_{j}$ is distributed according to (2.7). This expectation is a Selberg integral equal to [14, 17.6.2]

[TABLE]

and the result follows after some elementary algebra. ∎

We note that this result can be proved as a consequence of theorem 5 by Ismail [12]: for normalized Gegenbauer polynomials defined as

[TABLE]

the Turàn determinant equals

[TABLE]

Applying Nagel’s identity,

[TABLE]

so that

[TABLE]

Elementary algebra yields the result. We remark the similarity between the former formula and Nagel identity.

8. Conclusion

Some new results about Relativistic Hermite polynomials have been shown; the important fact is that several tools (subordination, moment representation) have been used, depending on the type of the result.