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A note on degenerate generalized Laguerre polynomials and Lah numbers
Advances in Difference Equations volume 2021, Article number: 421 (2021)
Abstract
The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers, including an explicit expression, a Rodrigues type formula, and expressions for the derivatives. The novelty of the present paper is that it is the first paper on degenerate versions of orthogonal polynomials.
1 Introduction
The generalized Laguerre polynomials are classical orthogonal polynomials which are orthogonal with respect to the gamma distribution \(e^{-x} x^{\alpha }\,dx\) on the interval \((0,\infty )\). The generalized Laguerre polynomials are widely used in many problems of quantum mechanics, mathematical physics and engineering. In quantum mechanics, the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a generalized Laguerre polynomial [14]. In mathematical physics, vibronic transitions in the Franck–Condon approximation can also be described by using Laguerre polynomials [6]. In engineering, the wave equation is solved for the time domain electric field integral equation for arbitrary shaped conducting structures by expressing the transient behaviors in terms of Laguerre polynomials [4].
The aim of this paper is to introduce the degenerate generalized Laguerre polynomials as the degenerate version of the generalized Laguerre polynomials and to derive some properties related to those polynomials and Lah numbers. In more detail, we obtain an explicit formula and a Rodrigues type formula for the degenerate Laguerre polynomials. We also get explicit expressions for the degenerate generalized Laguerre polynomial for \(\alpha =-1\), an identity involving Lah numbers, the falling factorial moment of the degenerate Poisson random variable with parameter α, and expressions for the derivatives of the degenerate generalized Laguerre polynomials.
We should mention here that degenerate versions of many special numbers and polynomials have been explored and many interesting results have been obtained in recent years [8, 11, 12]. Furthermore, these have been done not only for special numbers and polynomials but also for transcendental functions like gamma functions [10]. The novelty of the present paper is that this is the first paper which treats degenerate versions of orthogonal polynomials. For the rest of this section, we will recall some necessary facts that will be used throughout this paper.
The Laguerre polynomial \(L_{n}(x)\) satisfies the second-order linear differential equation
while the generalized Laguerre polynomial (or the associated Laguerre polynomial) \(L_{n}^{(\alpha )}(x)\) satisfies the second-order linear differential equation
The Rodrigues formula of the Laguerre polynomial \(L_{n}(x)\) is given by
while that of the generalized Laguerre polynomial \(L_{n}^{(\alpha )}(x)\) is given by
The generating function of generalized Laguerre polynomials is given by
From (3), we get
Note that
The rising factorial sequence is defined as
while the falling factorial sequence is defined as
We note that the Lah numbers are defined by
From (5), we can easily derive the following equation:
For any \(\lambda \in \mathbb{R}\), the degenerate exponential function is defined by
where \((x)_{0,\lambda }=1\), \((x)_{n,\lambda }=x(x-\lambda )\cdots (x-(n-1) \lambda )\), \((n\ge 1)\). For \(x=1\), we use the brief notation \(e_{\lambda }(t)=e_{\lambda }^{1}(t)\).
2 Degenerate generalized Laguerre polynomials
For any \(\alpha \in \mathbb{R}\), we consider the degenerate generalized Laguerre polynomials given by
From (7), we note that
Therefore, by (8) and (9), we obtain the following theorem.
Theorem 1
For \(n\ge 0\), we have
Now, by using Theorem 1, we observe that
Therefore, by (10), we obtain the following theorem.
Theorem 2
For \(n\ge 0\), we have
By using Leibniz rule and Theorem 1, we have
Thus, we obtain Rodrigues type formula for the degenerate generalized Laguerre polynomials.
Theorem 3
(Rodrigues type formula)
For \(n\ge 0\), we have
For \(\alpha =-1\), from Theorem 3, we have
On the other hand, by (8), we get
From (7), we can derive the following equation:
Thus, by (13) and (14), we get
where \(L(n,k)=\binom{n-1}{k-1}\frac{n!}{k!}\) is the Lah number.
Therefore, we obtain the following theorem.
Theorem 4
For \(n\ge 0\), we have
From Theorem 1, we note that
Thus, by (16), we get
In particular, \(\alpha =-1\), we have
Now, we observe that
Therefore, by (19), we obtain the following theorem.
Theorem 5
For \(n\ge 1\), we have
Since
we have the following corollary.
Corollary 6
For \(n\ge 1\), we have
3 Degenerate Poisson random variables
Let X be the Poisson random variable with parameter \(\alpha (>0)\). Then the probability mass function of X is given by
It is easy to show that
Thus, we note that
Let \(X_{\lambda }\) be the degenerate Poisson random variable with parameter \(\alpha (>0)\). Then the probability mass function of \(X_{\lambda }\) is given by
Then the following falling factorial moment is given by
Assume that \(X_{\lambda }\) is the Poisson random variable with parameter \(\frac{1}{\alpha }(>0)\). Then, by using (20), we obtain
4 Derivatives of degenerate Laguerre polynomials
Let us consider the sequence \(y_{n,\lambda }(x)\) which is given by
where \(A(t)\) is an invertible series.
Note that \(y_{0,\lambda }(x)=A(0)\) is a constant. We now set \(F_{\lambda }=F_{\lambda }(x,t)=A(t)e_{\lambda } (- \frac{x}{1-t}t )\).
From (21), we note that
By (22), we get
From (21) and (23), we can derive the following equation:
By comparing the coefficients on both sides of (24), we get
where \(y_{n,\lambda }^{\prime }(x)=\frac{d}{dx}y_{n,\lambda }(x)\).
Now, we observe that
From (22) and (26), we can derive the following equation:
Thus, by comparing the coefficients on both sides of (27), we get
Therefore, we obtain the following theorem.
Theorem 7
Let
where \(A(t)\) is an invertible series.
Then, for \(n\ge 1\), we have
and
where \(y_{n,\lambda }^{\prime }(x)=\frac{d}{dx}y_{n,\lambda }(x) \).
From the definition of the degenerate generalized Laguerre polynomials in (8), we observe that
In Theorem 7, let us take \(A(t)=(1-t)^{-\alpha -1}\). Then we have
Thus, we note that \(y_{n,\lambda }(x)=L_{n,\lambda }^{(\alpha )}(x)\), \((n\ge 0)\).
Therefore, by Theorem 7, (29), and (30), we obtain the following corollary.
Corollary 8
For \(n\ge 1\), we have the following derivative formulas:
Remark 9
The last derivative formula in Corollary 8 was drawn attention to by one of the referees to whom we thank.
5 Conclusion
In this paper, we introduced the degenerate generalized Laguerre polynomials, which are the first degenerate versions of the orthogonal polynomials, and derived some results related to those polynomials and Lah numbers. Some of the results are an explicit expression, Rodrigues type formula, and some expressions for the derivatives of the degenerate generalized Laguerre polynomials.
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Acknowledgements
We would like to thank the referees who helped improve the original manuscript in its present form.
Funding
This work was supported by the Basic Science Research Program, the National Research Foundation of Korea (NRF-2021R1F1A1050151).
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Kim, T., Dolgy, D.V., Kim, D.S. et al. A note on degenerate generalized Laguerre polynomials and Lah numbers. Adv Differ Equ 2021, 421 (2021). https://doi.org/10.1186/s13662-021-03574-8
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DOI: https://doi.org/10.1186/s13662-021-03574-8
MSC
- 11B83
- 42C05
- 60E99
Keywords
- Degenerate generalized Laguerre polynomials
- Lah numbers
- Degenerate exponential function