A note on degenerate generalized Laguerre polynomials and Lah numbers

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.


INTRODUCTION
The generalized Laguerre polynomials are classical orthogonal polynomials which are orthogonal with respect to the gamma distribution e −x x α dx on the interval (0, ∞). 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 α = −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 obtained in recent years [8,10,12]. Furthermore, these have been done not only for special numbers and polynomials but also for transcendental functions like gamma functions [11]. 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 Lagueree polynomial L n (x) satisfies the second-order linear differential equation while the generalized Laguerre polynomial (or the associated Laguerre polynomial) L (α) n (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 Lagurerre polynomial L (α) n (x) is given by [2,9,16,17]).
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 [3,5,8,13,15]).
For any λ ∈ R, the degenerate exponential function is defined by . For x = 1, we use the notation by e λ (t) = e 1 λ (t).

DEGENERATE GENERALIZED LAGUERRE POLYNOMIALS
For any α ∈ 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.
Now, by using Theorem1, we observe that Therefore, by (10), we obtain the following theorem.
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 ≥ 0, we have For α = −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) = n−1 k−1 n! k! is the Lah number. Therefore, we obtain the following theorem.
Thus, we have the following corollary.

DEGENERATE POISSON RANDOM VARIABLES
Let X be the Poisson random variable with parameter α(> 0). Then the probability mass function of X is given by It is easy to show that Thus, we note that Let X λ be the degenerate Poisson random variable with parameter α(> 0). Then the probability mass function of X λ is given by , (see [12]).
Then the following falling factorial moment is given by Assume that X λ is the Poisson random variable with parameter 1 α (> 0). Then, by using (20), we obtain d n dα n e λ

DERIVATIVES OF DEGENERATE LAGUERRE POLYNOMIALS
Let us consider the sequence y n,λ (x) which are given by where A(t) is an invertible series.
Note that y 0,λ (x) = A(0) is a constant. We now set 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,λ (x) = d dx y n,λ (x). Now, we observe that From (22) and (26), we can derive the following equation (27) Thus, by comparing the coefficients on both sides of (27), we get Therefore, we obtain the following theorem.

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.
As one of our future projects, we would like to continue to study degenerate versions of many special numbers and polynomials.