1 Introduction

The Laguerre polynomials, \(L_{n} (x)\) (\(n\geq0\)), are defined by the generating function

$$ \frac{ e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty} L_{n}(x)t^{n} \quad (\mbox{see [1, 2]}). $$
(1)

Indeed, the Laguerre polynomial \(L_{n}(x)\) is a solution of the second order linear differential equation

$$ xy''+(1-x)y+ny=0\quad (\mbox{see [2--5]}). $$
(2)

From (1), we can get the following equation:

$$\begin{aligned} \sum_{n=0}^{\infty} L_{n}(x)t^{n} =&\frac{ e^{-\frac{xt}{1-t}}}{1-t}= \sum _{m=0}^{\infty} \frac{(-1)^{m} x^{m} t^{m}}{m!}(1-t)^{-m-1} \\ =& \sum_{m=0}^{\infty} \frac{(-1)^{m} x^{m} t^{m}}{m!}\sum _{l=0}^{\infty}\binom{m+l}{l}t^{l} \\ =&\sum_{n=0}^{\infty} \Biggl(\sum _{m=0}^{n}\frac{(-1)^{m} \binom{n}{m}x^{m} }{m!}\Biggr)t^{n}. \end{aligned}$$
(3)

Thus by (3), we get immediately the following equation:

$$ L_{n}(x)= \sum_{m=0}^{n} \frac{(-1)^{m} \binom{n}{m}x^{m} }{m!}\quad (n\geq0)\ \bigl(\mbox{see [2, 6--8]} \bigr). $$
(4)

Alternatively, the Laguerre polynomials are also defined by the recurrence relation as follows:

$$ \begin{aligned} & L_{0}(x)=1,\qquad L_{1}(x)=1-x, \\ & (n+1)L_{n+1}(x)=(2n+1-x)L_{n}(x)-nL_{n-1}(x)\quad (n \geq1). \end{aligned} $$
(5)

The Rodrigues’ formula for the Laguerre polynomials is given by

$$ L_{n}(x)= \frac{1}{n!}e^{x} \frac{d^{n}}{dx^{n}} \bigl(e^{-x}x^{n}\bigr)\quad (n\geq0). $$
(6)

The first few of \(L_{n} (x)\) (\(n\geq0\)) are

$$\begin{aligned} &L_{0}(x)= 1, \\ &L_{1}(x)= 1-x, \\ &L_{2}(x)= \frac{1}{2}\bigl(x^{2}-4x+2\bigr), \\ &L_{3}(x)= \frac{1}{6}\bigl(-x^{3} + 9x^{2}-18x+6\bigr), \\ &L_{4}(x)=\frac{1}{24}\bigl(x^{4} -16x^{3} + 72x^{2}-96x+24\bigr). \end{aligned} $$

The Laguerre polynomials arise from quantum mechanics in the radial part of the solution of the Schrödinger equation for a one-electron action. They also describe the static Wigner functions of oscillator system in the quantum mechanics of phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator (see [4, 5, 9]). A contour integral that is commonly taken as the definition of the Laguerre polynomial is given by

$$ L_{n}(x)= \frac{1}{2\pi i} \oint_{C} \frac{e^{\frac{-xt}{1-t}}}{1-t}t^{-n-1}\,dt\quad \bigl(\mbox{see [4, 5, 10, 11]} \bigr), $$
(7)

where the contour encloses the origin but not the point \(z=1\).

FDEs (fractional differential equations) have wide applications in such diverse areas as fluid mechanics, plasma physics, dynamical processes and finance, etc. Most FDEs do not have exact solutions and hence numerical approximation techniques must be used. Spectral methods are widely used to numerically solve various types of integral and differential equations due to their high accuracy and employ orthogonal systems as basis functions. It is remarkable that a new family of generalized Laguerre polynomials are introduced in applying spectral methods for numerical treatments of FDEs in unbounded domains. They can also be used in solving some differential equations (see [12–17]).

Also, it should be mentioned that the modified generalized Laguerre operational matrix of fractional integration is applied in order to solve linear multi-order FDEs which are important in mathematical physics (see [12–17]).

Many authors have studied the Laguerre polynomials in mathematical physics, combinatorics and special functions (see [1–30]). For the applications of special functions and polynomials, one may referred to the papers (see [18, 19, 28]).

In [22], Kim studied nonlinear differential equations arising from Frobenius-Euler polynomials and gave some interesting identities. In this paper, we derive a family of ordinary differential equations from the generating function of the Laguerre polynomials. Then these differential equations are used in order to obtain some properties and new identities for those polynomials.

2 Laguerre polynomials arising from linear differential equations

Let

$$ F= F(t,x)=\frac{1}{1-t}e^{\frac{-xt}{1-t}}. $$
(8)

From (8), we note that

$$ F^{(1)}= \frac{dF(t,x)}{dt}=\bigl((1-t)^{-1}-x(1-t)^{-2} \bigr)F. $$
(9)

Thus, by (3), we get

$$ F^{(2)}= \frac{dF^{(1)}}{dt}=\bigl(2(1-t)^{-2}-4x(1-t)^{-3}+ x^{2}(1-t)^{-4}\bigr)F $$
(10)

and

$$ F^{(3)}= \frac{dF^{(2)}}{dt}=\bigl(6(1-t)^{-3}-18x(1-t)^{-4}+ 9x^{2}(1-t)^{-5}-x^{3}(1-t)^{-6} \bigr)F. $$
(11)

So we are led to put

$$ F^{(N)}= \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)F, $$
(12)

where \(N=0, 1, 2, \ldots \) .

From (12), we can get equation (13):

$$\begin{aligned} F^{(N+1)} =& \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x)i(1-t)^{-i-1} \Biggr)F + \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)F^{(1)} \\ =& \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x)i(1-t)^{-i-1} \Biggr)F \\ &{}+ \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)\bigl((1-t)^{-1}-x(1-t)^{-2} \bigr)F \\ = &\Biggl(\sum_{i=N}^{2N}(i+1)a_{i-N}(N,x) (1-t)^{-i-1}- x\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i-2} \Biggr)F \\ =& \Biggl(\sum_{i=N+1}^{2N+1}i a_{i-N-1}(N,x) (1-t)^{-i}- x\sum_{i=N+2}^{2N+2}a_{i-N-2}(N,x) (1-t)^{-i} \Biggr)F. \end{aligned}$$
(13)

Replacing N by \(N+1\) in (12), we get

$$ F^{(N+1)} = \Biggl(\sum_{i=N+1}^{2N+2}a_{i-N-1}(N+1,x) (1-t)^{-i} \Biggr)F. $$
(14)

Comparing the coefficients on both sides of (13) and (14), we have

$$\begin{aligned}& a_{0}(N+1,x) =(N+1) a_{0}(N,x), \end{aligned}$$
(15)
$$\begin{aligned}& a_{N+1}(N+1,x) =-x a_{N}(N,x), \end{aligned}$$
(16)

and

$$ a_{i-N-1}(N+1,x) =ia_{i-N-1}(N,x) -x a_{i-N-2}(N,x) \quad (N+2 \leq i \leq2N+1). $$
(17)

We note that

$$ F= F^{(0)}= a_{0}(0,x)F . $$
(18)

Thus, by (18), we get

$$ a_{0}(0,x)=1. $$
(19)

From (9) and (12), we note that

$$ \bigl((1-t)^{-1}-x(1-t)^{-2}\bigr)F= F^{(1)}=\bigl( a_{0}(1,x) (1-t)^{-1}+ a_{1}(1,x) (1-t)^{-2}\bigr)F. $$
(20)

Thus, by comparing the coefficients on both sides of (20), we get

$$ a_{0}(1,x)=1,\qquad a_{1}(1,x)=-x. $$
(21)

From (15), (16), we get

$$\begin{aligned} a_{0}(N+1,x) =&(N+1)a_{N}(N,x)=(N+1)N a_{N-1}(N-1,x) \cdots \\ =&(N+1)N(N-1) \cdots2a_{0}(1,x)=(N+1)! \end{aligned}$$
(22)

and

$$\begin{aligned} a_{N+1}(N+1,x) =&(-x)a_{N}(N,x)=(-x)^{2} a_{N-1}(N-1,x) \cdots \\ =&(-x)^{N} a_{1}(1,x)=(-x)^{N+1} . \end{aligned}$$
(23)

We observe that the matrix \([a_{i} (j, x) ]_{0\leq i,j \leq N} \) is given by

$$\left[ \begin{matrix} 1 & 1! & 2! \cdots& N! \\ 0 & (-x) &\cdots \\ 0 & 0 &(-x)^{2} \\ \vdots &\vdots \\ 0 & 0& \cdots& (-x)^{N} \end{matrix}\right]. $$

From (17), we can get the following equations:

$$\begin{aligned}& \begin{aligned}[b] a_{1} (N+1,x)&=-x a_{0}(N,x)+(N+2)a_{1}(N,x) \\ &=-x \bigl\{ a_{0}(N,x)+(N+2)a_{0}(N-1,x)\bigr\} +(N+2) (N+1) a_{1}(N-1,x) \\ &=\cdots \\ & =-x \sum_{i=0}^{N-1}(N+2)_{i} a_{0}(N-i,x)+ (N+2) (N+1) \cdots 3a_{1}(1,x) \\ & =-x \sum_{i=0}^{N-1}(N+2)_{i} a_{0}(N-i,x)+ (N+2) (N+1) \cdots3(-x) \\ &=-x \sum_{i=0}^{N}(N+2)_{i} a_{0}(N-i,x), \end{aligned} \end{aligned}$$
(24)
$$\begin{aligned}& \begin{aligned}[b] a_{2} (N+1,x)&=-x a_{1}(N,x)+(N+3)a_{2}(N,x) \\ &=-x \bigl\{ a_{1}(N,x)+(N+3)a_{1}(N-1,x) \bigr\} +(N+3) (N+2) a_{2}(N-1,x) \\ & =\cdots \\ & =-x \sum_{i=0}^{N-2}(N+3)_{i} a_{1}(N-i,x)+ (N+3) (N+2) \cdots 5a_{2}(2,x) \\ &=-x \sum_{i=0}^{N-2}(N+3)_{i} a_{1}(N-i,x)+(N+3) (N+2) \cdots5(-x)^{2} \\ &=-x \sum_{i=0}^{N-1}(N+3)_{i} a_{1}(N-i,x), \end{aligned} \end{aligned}$$
(25)

and

$$\begin{aligned}& \begin{aligned}[b] a_{3} (N+1,x)&=-x a_{2}(N,x)+(N+4)a_{3}(N,x) \\ &=-x \bigl\{ a_{2}(N,x)+(N+4)a_{2}(N-1,x) \bigr\} +(N+4) (N+3) a_{3}(N-1,x) \\ & =\cdots \\ & =-x \sum_{i=0}^{N-3}(N+4)_{i} a_{2}(N-i,x)+ (N+4) (N+3) \cdots 7a_{3}(3,x) \\ &=-x\sum_{i=0}^{N-3}(N+4)_{i} a_{2}(N-i,x)+ (N+4) (N+3) \cdots7(-x)^{3} \\ &=-x \sum_{i=0}^{N-2}(N+4)_{i} a_{2}(N-i,x), \end{aligned} \end{aligned}$$
(26)

where \((x)_{n} =x(x-1) \cdots(x-n+1)\) (\(n\geq1\)), and \((x)_{0} =1\).

Continuing this process, we have

$$ a_{j} (N+1,x)=-x \sum_{i=0}^{N-j+1}(N+j+1)_{i} a_{j-1}(N-i,x), $$
(27)

where \(j=1,2, \ldots, N\). Now we give explicit expressions for \(a_{j} (N+1,x)\), \(j=1,2, \ldots, N\). From (22) and (24), we note that

$$\begin{aligned}& \begin{aligned}[b] a_{1} (N+1,x) &=-x \sum_{i_{1}=0}^{N}(N+2)_{i_{1}} a_{0}(N-i_{1},x) \\ &=-x \sum_{i_{1}=0}^{N}(N+2)_{i_{1}}( N-i_{1})!. \end{aligned} \end{aligned}$$
(28)

By (25) and (28), we get

$$\begin{aligned}& \begin{aligned}[b] a_{2} (N+1,x) &=-x \sum_{i_{2}=0}^{N-1}(N+3)_{i_{2}} a_{1}(N-i_{2},x) \\ &=(-x)^{-2}\sum_{i_{2}=0}^{N-1}\sum _{i_{1}=0}^{N-i_{2}-1}(N+3)_{i_{2}}( N-i_{2}+1)_{i_{1}}( N-i_{2}-i_{1}-1)!. \end{aligned} \end{aligned}$$
(29)

From (26) and (29), we get

$$\begin{aligned}& \begin{aligned}[b] a_{3} (N+1,x) ={}&{-}x \sum_{i_{3}=0}^{N-2}(N+4)_{i_{3}} a_{2}(N-i_{3},x) \\ ={}&(-x)^{-3}\sum_{i_{3}=0}^{N-2}\sum _{i_{2}=0}^{N-i_{3}-2}\sum_{i_{1}=0}^{N-i_{3}-i_{2}-2}(N+4)_{i_{3}}( N-i_{3}+2)_{i_{2}}( N-i_{3}-i_{2})_{i_{1}} \\ &{}\times ( N-i_{3}-i_{2}-i_{1}-2)!. \end{aligned} \end{aligned}$$
(30)

By continuing this process, we get

$$\begin{aligned}& \begin{aligned}[b] a_{j} (N+1,x)={}&(-x)^{j}\sum _{i_{j}=0}^{N-j+1}\sum_{i_{j-1}=0}^{N-i_{j}-j+1} \cdots\sum_{i_{1}=0}^{N-i_{j}- \cdots -i_{2}-j+1}(N+j+1)_{i_{j}} \\ &{}\times\Biggl(\prod_{k=2}^{j} N-i_{j}- \cdots-i_{k}-\bigl(j-(2k-1)\bigr)_{i_{k-1}} \Biggr) \\ &{}\times ( N-i_{j}- \cdots-i_{1}-j+1)!. \end{aligned} \end{aligned}$$
(31)

Therefore, we obtain the following theorem.

Theorem 1

The linear differential equation

$$F^{(N)}= \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)F \quad (N\in\mathbb{N}) $$

has a solution \(F=F(t,x)=(1-t)^{-1}\exp(-\frac{xt}{1-t})\), where \(a_{0}(N,x)=N!\), \(a_{N}(N,x)=(-x)^{N}\),

$$\begin{aligned}& \begin{aligned}[b] a_{j} (N,x)={}&(-x)^{j}\sum _{i_{j}=0}^{N-j}\sum_{i_{j-1}=0}^{N-i_{j}-j} \cdots\sum_{i_{1}=0}^{N-i_{j}- \cdots-i_{2}-j}(N+j)_{i_{j}} \\ &{}\times\Biggl(\prod_{k=2}^{j}\bigl( N-i_{j}- \cdots-i_{k}-\bigl(j-(2k-2)\bigr) \bigr)_{i_{k-1}} \Biggr) ( N-i_{j}- \cdots-i_{1}-j)!. \end{aligned} \end{aligned}$$

From (1), we note that

$$ F=F(t,x)=\frac{ e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty} L_{n}(x)t^{n}. $$
(32)

Thus, by (32), we get

$$ F^{(N)}=\biggl(\frac{d}{dt}\biggr)^{N}F(t,x)=\sum _{n=N}^{\infty} L_{n}(x) (n)_{N}t^{n-N}=\sum_{n=0} ^{\infty} L_{n+N}(x) (n+N)_{N}t^{n}. $$
(33)

On the other hand, by Theorem 1, we have

$$\begin{aligned}& \begin{aligned}[b] F^{(N)}&= \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)F \\ &=\sum_{i=N}^{2N}a_{i-N}(N,x)\sum _{l=0}^{\infty}{i+l-1\choose l}t^{l} \sum_{k=0}^{\infty}L_{k}(x)t^{k} \\ &=\sum_{i=N}^{2N}a_{i-N}(N,x)\sum _{n=0}^{\infty} \Biggl(\sum _{l=0}^{n}{i+l-1\choose l}L_{n-l}(x) \Biggr)t^{n} \\ &=\sum_{n=0}^{\infty} \Biggl(\sum _{i=N}^{2N}a_{i-N}(N,x) \sum _{l=0}^{N}{i+l-1\choose l}L_{n-l}(x) \Biggr) t^{n}. \end{aligned} \end{aligned}$$
(34)

Therefore, by comparing the coefficients on both sides of (33) and (34), we have the following theorem.

Theorem 2

For \(n \in\mathbb{N} \cup\{0\}\) and \(N \in\mathbb{N} \), we have

$$L_{n+N}(x)=\frac{1}{(n+N)_{N}}\sum_{i=N}^{2N}a_{i-N}(N,x) \sum_{l=0}^{N}{i+l-1\choose l}L_{n-l}(x), $$

where \(a_{0}(N,x)=N!\), \(a_{N}(N,x)=(-x)^{N}\),

$$\begin{aligned}& \begin{aligned}[b] a_{j} (N,x)={}&(-x)^{j}\sum _{i_{j}=0}^{N-j}\sum_{i_{j-1}=0}^{N-i_{j}-j} \cdots\sum_{i_{1}=0}^{N-i_{j}- \cdots-i_{2}-j}(N+j)_{i_{j}} \\ &{}\times\Biggl(\prod_{k=2}^{j}\bigl( N-i_{j}- \cdots-i_{k}-\bigl(j-(2k-2)\bigr) \bigr)_{i_{k-1}} \Biggr) ( N-i_{j}- \cdots-i_{1}-j)!. \end{aligned} \end{aligned}$$

3 Conclusion

It has been demonstrated that it is a fascinating idea to use differential equations associated with the generating function (or a slight variant of generating function) of special polynomials or numbers. Immediate applications of them have been in deriving interesting identities for the special polynomials or numbers. Along this line of research, here we derived a family of differential equations from the generating function of the Laguerre polynomials. Then from these differential equations we obtained interesting new identities for those polynomials.