Skip to content


  • Research
  • Open Access

Some identities of Laguerre polynomials arising from differential equations

Advances in Difference Equations20162016:159

  • Received: 25 January 2016
  • Accepted: 12 June 2016
  • Published:


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.


  • Laguerre polynomials
  • differential equations


  • 05A19
  • 33C45
  • 11B37
  • 35G35

1 Introduction

The Laguerre polynomials, \(L_{n} (x)\) (\(n\geq0\)), are defined by the generating function
$$ \frac{ e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty} L_{n}(x)t^{n} \quad (\mbox{see [1, 2]}). $$
Indeed, the Laguerre polynomial \(L_{n}(x)\) is a solution of the second order linear differential equation
$$ xy''+(1-x)y+ny=0\quad (\mbox{see [2--5]}). $$
From (1), we can get the following equation:
$$\begin{aligned} \sum_{n=0}^{\infty} L_{n}(x)t^{n} =&\frac{ e^{-\frac{xt}{1-t}}}{1-t}= \sum _{m=0}^{\infty} \frac{(-1)^{m} x^{m} t^{m}}{m!}(1-t)^{-m-1} \\ =& \sum_{m=0}^{\infty} \frac{(-1)^{m} x^{m} t^{m}}{m!}\sum _{l=0}^{\infty}\binom{m+l}{l}t^{l} \\ =&\sum_{n=0}^{\infty} \Biggl(\sum _{m=0}^{n}\frac{(-1)^{m} \binom{n}{m}x^{m} }{m!}\Biggr)t^{n}. \end{aligned}$$
Thus by (3), we get immediately the following equation:
$$ L_{n}(x)= \sum_{m=0}^{n} \frac{(-1)^{m} \binom{n}{m}x^{m} }{m!}\quad (n\geq0)\ \bigl(\mbox{see [2, 6--8]} \bigr). $$
Alternatively, the Laguerre polynomials are also defined by the recurrence relation as follows:
$$ \begin{aligned} & L_{0}(x)=1,\qquad L_{1}(x)=1-x, \\ & (n+1)L_{n+1}(x)=(2n+1-x)L_{n}(x)-nL_{n-1}(x)\quad (n \geq1). \end{aligned} $$
The Rodrigues’ formula for the Laguerre polynomials is given by
$$ L_{n}(x)= \frac{1}{n!}e^{x} \frac{d^{n}}{dx^{n}} \bigl(e^{-x}x^{n}\bigr)\quad (n\geq0). $$
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
$$ L_{n}(x)= \frac{1}{2\pi i} \oint_{C} \frac{e^{\frac{-xt}{1-t}}}{1-t}t^{-n-1}\,dt\quad \bigl(\mbox{see [4, 5, 10, 11]} \bigr), $$
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 [1217]).

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 [1217]).

Many authors have studied the Laguerre polynomials in mathematical physics, combinatorics and special functions (see [130]). 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

$$ F= F(t,x)=\frac{1}{1-t}e^{\frac{-xt}{1-t}}. $$
From (8), we note that
$$ F^{(1)}= \frac{dF(t,x)}{dt}=\bigl((1-t)^{-1}-x(1-t)^{-2} \bigr)F. $$
Thus, by (3), we get
$$ F^{(2)}= \frac{dF^{(1)}}{dt}=\bigl(2(1-t)^{-2}-4x(1-t)^{-3}+ x^{2}(1-t)^{-4}\bigr)F $$
$$ F^{(3)}= \frac{dF^{(2)}}{dt}=\bigl(6(1-t)^{-3}-18x(1-t)^{-4}+ 9x^{2}(1-t)^{-5}-x^{3}(1-t)^{-6} \bigr)F. $$
So we are led to put
$$ F^{(N)}= \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)F, $$
where \(N=0, 1, 2, \ldots \) .
From (12), we can get equation (13):
$$\begin{aligned} F^{(N+1)} =& \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x)i(1-t)^{-i-1} \Biggr)F + \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)F^{(1)} \\ =& \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x)i(1-t)^{-i-1} \Biggr)F \\ &{}+ \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)\bigl((1-t)^{-1}-x(1-t)^{-2} \bigr)F \\ = &\Biggl(\sum_{i=N}^{2N}(i+1)a_{i-N}(N,x) (1-t)^{-i-1}- x\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i-2} \Biggr)F \\ =& \Biggl(\sum_{i=N+1}^{2N+1}i a_{i-N-1}(N,x) (1-t)^{-i}- x\sum_{i=N+2}^{2N+2}a_{i-N-2}(N,x) (1-t)^{-i} \Biggr)F. \end{aligned}$$
Replacing N by \(N+1\) in (12), we get
$$ F^{(N+1)} = \Biggl(\sum_{i=N+1}^{2N+2}a_{i-N-1}(N+1,x) (1-t)^{-i} \Biggr)F. $$
Comparing the coefficients on both sides of (13) and (14), we have
$$\begin{aligned}& a_{0}(N+1,x) =(N+1) a_{0}(N,x), \end{aligned}$$
$$\begin{aligned}& a_{N+1}(N+1,x) =-x a_{N}(N,x), \end{aligned}$$
$$ a_{i-N-1}(N+1,x) =ia_{i-N-1}(N,x) -x a_{i-N-2}(N,x) \quad (N+2 \leq i \leq2N+1). $$
We note that
$$ F= F^{(0)}= a_{0}(0,x)F . $$
Thus, by (18), we get
$$ a_{0}(0,x)=1. $$
From (9) and (12), we note that
$$ \bigl((1-t)^{-1}-x(1-t)^{-2}\bigr)F= F^{(1)}=\bigl( a_{0}(1,x) (1-t)^{-1}+ a_{1}(1,x) (1-t)^{-2}\bigr)F. $$
Thus, by comparing the coefficients on both sides of (20), we get
$$ a_{0}(1,x)=1,\qquad a_{1}(1,x)=-x. $$
From (15), (16), we get
$$\begin{aligned} a_{0}(N+1,x) =&(N+1)a_{N}(N,x)=(N+1)N a_{N-1}(N-1,x) \cdots \\ =&(N+1)N(N-1) \cdots2a_{0}(1,x)=(N+1)! \end{aligned}$$
$$\begin{aligned} a_{N+1}(N+1,x) =&(-x)a_{N}(N,x)=(-x)^{2} a_{N-1}(N-1,x) \cdots \\ =&(-x)^{N} a_{1}(1,x)=(-x)^{N+1} . \end{aligned}$$
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:
$$\begin{aligned}& \begin{aligned}[b] a_{1} (N+1,x)&=-x a_{0}(N,x)+(N+2)a_{1}(N,x) \\ &=-x \bigl\{ a_{0}(N,x)+(N+2)a_{0}(N-1,x)\bigr\} +(N+2) (N+1) a_{1}(N-1,x) \\ &=\cdots \\ & =-x \sum_{i=0}^{N-1}(N+2)_{i} a_{0}(N-i,x)+ (N+2) (N+1) \cdots 3a_{1}(1,x) \\ & =-x \sum_{i=0}^{N-1}(N+2)_{i} a_{0}(N-i,x)+ (N+2) (N+1) \cdots3(-x) \\ &=-x \sum_{i=0}^{N}(N+2)_{i} a_{0}(N-i,x), \end{aligned} \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] a_{2} (N+1,x)&=-x a_{1}(N,x)+(N+3)a_{2}(N,x) \\ &=-x \bigl\{ a_{1}(N,x)+(N+3)a_{1}(N-1,x) \bigr\} +(N+3) (N+2) a_{2}(N-1,x) \\ & =\cdots \\ & =-x \sum_{i=0}^{N-2}(N+3)_{i} a_{1}(N-i,x)+ (N+3) (N+2) \cdots 5a_{2}(2,x) \\ &=-x \sum_{i=0}^{N-2}(N+3)_{i} a_{1}(N-i,x)+(N+3) (N+2) \cdots5(-x)^{2} \\ &=-x \sum_{i=0}^{N-1}(N+3)_{i} a_{1}(N-i,x), \end{aligned} \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b] a_{3} (N+1,x)&=-x a_{2}(N,x)+(N+4)a_{3}(N,x) \\ &=-x \bigl\{ a_{2}(N,x)+(N+4)a_{2}(N-1,x) \bigr\} +(N+4) (N+3) a_{3}(N-1,x) \\ & =\cdots \\ & =-x \sum_{i=0}^{N-3}(N+4)_{i} a_{2}(N-i,x)+ (N+4) (N+3) \cdots 7a_{3}(3,x) \\ &=-x\sum_{i=0}^{N-3}(N+4)_{i} a_{2}(N-i,x)+ (N+4) (N+3) \cdots7(-x)^{3} \\ &=-x \sum_{i=0}^{N-2}(N+4)_{i} a_{2}(N-i,x), \end{aligned} \end{aligned}$$
where \((x)_{n} =x(x-1) \cdots(x-n+1)\) (\(n\geq1\)), and \((x)_{0} =1\).
Continuing this process, we have
$$ a_{j} (N+1,x)=-x \sum_{i=0}^{N-j+1}(N+j+1)_{i} a_{j-1}(N-i,x), $$
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
$$\begin{aligned}& \begin{aligned}[b] a_{1} (N+1,x) &=-x \sum_{i_{1}=0}^{N}(N+2)_{i_{1}} a_{0}(N-i_{1},x) \\ &=-x \sum_{i_{1}=0}^{N}(N+2)_{i_{1}}( N-i_{1})!. \end{aligned} \end{aligned}$$
By (25) and (28), we get
$$\begin{aligned}& \begin{aligned}[b] a_{2} (N+1,x) &=-x \sum_{i_{2}=0}^{N-1}(N+3)_{i_{2}} a_{1}(N-i_{2},x) \\ &=(-x)^{-2}\sum_{i_{2}=0}^{N-1}\sum _{i_{1}=0}^{N-i_{2}-1}(N+3)_{i_{2}}( N-i_{2}+1)_{i_{1}}( N-i_{2}-i_{1}-1)!. \end{aligned} \end{aligned}$$
From (26) and (29), we get
$$\begin{aligned}& \begin{aligned}[b] a_{3} (N+1,x) ={}&{-}x \sum_{i_{3}=0}^{N-2}(N+4)_{i_{3}} a_{2}(N-i_{3},x) \\ ={}&(-x)^{-3}\sum_{i_{3}=0}^{N-2}\sum _{i_{2}=0}^{N-i_{3}-2}\sum_{i_{1}=0}^{N-i_{3}-i_{2}-2}(N+4)_{i_{3}}( N-i_{3}+2)_{i_{2}}( N-i_{3}-i_{2})_{i_{1}} \\ &{}\times ( N-i_{3}-i_{2}-i_{1}-2)!. \end{aligned} \end{aligned}$$
By continuing this process, we get
$$\begin{aligned}& \begin{aligned}[b] a_{j} (N+1,x)={}&(-x)^{j}\sum _{i_{j}=0}^{N-j+1}\sum_{i_{j-1}=0}^{N-i_{j}-j+1} \cdots\sum_{i_{1}=0}^{N-i_{j}- \cdots -i_{2}-j+1}(N+j+1)_{i_{j}} \\ &{}\times\Biggl(\prod_{k=2}^{j} N-i_{j}- \cdots-i_{k}-\bigl(j-(2k-1)\bigr)_{i_{k-1}} \Biggr) \\ &{}\times ( N-i_{j}- \cdots-i_{1}-j+1)!. \end{aligned} \end{aligned}$$
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
$$ F=F(t,x)=\frac{ e^{-\frac{xt}{1-t}}}{1-t}=\sum_{n=0}^{\infty} L_{n}(x)t^{n}. $$
Thus, by (32), we get
$$ F^{(N)}=\biggl(\frac{d}{dt}\biggr)^{N}F(t,x)=\sum _{n=N}^{\infty} L_{n}(x) (n)_{N}t^{n-N}=\sum_{n=0} ^{\infty} L_{n+N}(x) (n+N)_{N}t^{n}. $$
On the other hand, by Theorem 1, we have
$$\begin{aligned}& \begin{aligned}[b] F^{(N)}&= \Biggl(\sum_{i=N}^{2N}a_{i-N}(N,x) (1-t)^{-i} \Biggr)F \\ &=\sum_{i=N}^{2N}a_{i-N}(N,x)\sum _{l=0}^{\infty}{i+l-1\choose l}t^{l} \sum_{k=0}^{\infty}L_{k}(x)t^{k} \\ &=\sum_{i=N}^{2N}a_{i-N}(N,x)\sum _{n=0}^{\infty} \Biggl(\sum _{l=0}^{n}{i+l-1\choose l}L_{n-l}(x) \Biggr)t^{n} \\ &=\sum_{n=0}^{\infty} \Biggl(\sum _{i=N}^{2N}a_{i-N}(N,x) \sum _{l=0}^{N}{i+l-1\choose l}L_{n-l}(x) \Biggr) t^{n}. \end{aligned} \end{aligned}$$
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.



This work was supported by the Dong-A university research fund. The first author is appointed as a chair professor at Tianjin Polytechnic University by Tianjin City in China from August 2015 to August 2019.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin, 300387, China
Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea
Department of Mathematics, Sogang University, Seoul, 04107, Republic of Korea
Department of Mathematics, Dong-A University, Busan, 49315, Republic of Korea
Department of Applied mathematics, Pukyong National University, Busan, 48513, Republic of Korea


  1. Kim, T: Identities involving Laguerre polynomials derived from umbral calculus. Russ. J. Math. Phys. 21(1), 36-45 (2014) MathSciNetView ArticleMATHGoogle Scholar
  2. Zill, DG, Cullen, MR: Advanced Engineering Mathematics. Jones & Bartlett, Boston (2005) MATHGoogle Scholar
  3. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. U.S. Government Printing Office, Washington (1964) MATHGoogle Scholar
  4. Arfken, G, Weber, H: Mathematical Methods for Physicists. Academic Press, San Diego (2000) MATHGoogle Scholar
  5. Bhrawy, AH, Alghamdi, MA: The operational matrix of Caputo fractional derivatives of modified generalized Laguerre polynomials and its applications. Adv. Differ. Equ. 2013, Article ID 307 (2013) MathSciNetView ArticleGoogle Scholar
  6. Srivastava, HM, Lin, S-D, Liu, S-J, Lu, H-C: Integral representations for the Lagrange polynomials, Shively’s pseudo-Laguerre polynomials, and the generalized Bessel polynomials. Russ. J. Math. Phys. 19(1), 121-130 (2012) MathSciNetView ArticleMATHGoogle Scholar
  7. Uspensky, JV: On the development of arbitrary functions in series of Hermite’s and Laguerre’s polynomials. Ann. Math. (2) 28(1-4), 593-619 (1926/1927) Google Scholar
  8. Watson, GN: An integral equation for the square of a Laguerre polynomial. J. Lond. Math. Soc. S1-11(4), 256 (1936) MathSciNetView ArticleMATHGoogle Scholar
  9. Karaseva, IA: Fast calculation of signal delay in RC-circuits based on Laguerre functions. Russ. J. Numer. Anal. Math. Model. 26(3), 295-301 (2011) MathSciNetView ArticleMATHGoogle Scholar
  10. Carlitz, L: Some generating functions for Laguerre polynomials. Duke Math. J. 35, 825-827 (1968) MathSciNetView ArticleMATHGoogle Scholar
  11. Carlitz, L: The product of several Hermite or Laguerre polynomials. Monatshefte Math. 66, 393-396 (1962) MathSciNetView ArticleMATHGoogle Scholar
  12. Baleanu, D, Bhrawy, AH, Taha, TM: Two efficient generalized Laguerre spectral algorithms for fractional initial value problems. Abstr. Appl. Anal. 2013, Article ID 546502 (2013) MathSciNetMATHGoogle Scholar
  13. Bhrawy, AH, Abdelkawy, MA, Alzahrani, AA, Baleanu, D, Alzahrani, EO: A Chebyshev-Laguerre-Gauss-Radau collocation scheme for solving a time fractional sub-diffusion equation on a semi-infinite domain. Proc. Rom. Acad., Ser. A : Math. Phys. Tech. Sci. Inf. Sci. 16, 490-498 (2015) MathSciNetGoogle Scholar
  14. Bhrawy, AH, Alhamed, YA, Baleanu, D, Al-Zahrani, AA: New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract. Calc. Appl. Anal. 17, 1137-1157 (2014) MathSciNetView ArticleMATHGoogle Scholar
  15. Bhrawy, AH, Alghamdi, MM, Taha, TM: A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line. Adv. Differ. Equ. 2012, Article ID 179 (2012) MathSciNetView ArticleGoogle Scholar
  16. Bhrawy, AH, Hafez, RM, Alzahrani, EO, Baleanu, D, Alzahrani, AA: Generalized Laguerre-Gauss-Radau scheme for first order hyperbolic equations on semi-infinite domains. Rom. J. Phys. 60, 918-934 (2015) Google Scholar
  17. Bhrawy, AH, Taha, TM, Alzahrani, EO, Baleanu, D, Alzahrani, AA: New operational matrices for solving fractional differential equations on the half-line. PLoS ONE 10(5), e0126620 (2015). doi:10.1371/journal.pone.0126620 View ArticleGoogle Scholar
  18. Chaurasia, VBL, Kumar, D: On the solutions of integral equations of Fredholm type with special functions. Tamsui Oxf. J. Inf. Math. Sci. 28, 49-61 (2012) MathSciNetMATHGoogle Scholar
  19. Chaurasia, VBL, Kumar, D: The integration of certain product involving special functions. Scientia, Ser. A, Math. Sci. 19, 7-12 (2010) MathSciNetMATHGoogle Scholar
  20. Chen, Y, Griffin, J: Deformed \(q^{-1}\)-Laguerre polynomials, recurrence coefficients, and non-linear difference equations. Acta Phys. Pol. A 46(9), 1871-1881 (2015) MathSciNetView ArticleGoogle Scholar
  21. Hegazi, AS, Mansour, M: Generalized q-modified Laguerre functions. Int. J. Theor. Phys. 41(9), 1803-1813 (2002) MathSciNetView ArticleMATHGoogle Scholar
  22. Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 132(12), 2854-2865 (2012) MathSciNetView ArticleMATHGoogle Scholar
  23. Kim, T, Kim, DS: Extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials. Abstr. Appl. Anal. 2012, Article ID 957350 (2012) MathSciNetMATHGoogle Scholar
  24. Kim, T, Rim, S-H, Dolgy, DV, Lee, S-H: Some identities on Bernoulli and Euler polynomials arising from the orthogonality of Laguerre polynomials. Adv. Differ. Equ. 2012, Article ID 201 (2012) MathSciNetView ArticleGoogle Scholar
  25. Koepf, W: Identities for families of orthogonal polynomials and special functions. Integral Transforms Spec. Funct. 5, 69-102 (1997) MathSciNetView ArticleMATHGoogle Scholar
  26. Filipuk, G, Smet, C: On the recurrence coefficients for generalized q-Laguerre polynomials. J. Nonlinear Math. Phys. 20(Suppl. 1), 48-56 (2013) MathSciNetView ArticleMATHGoogle Scholar
  27. Molano, LAM: An electrostatic model for zeros of classical Laguerre polynomials perturbed by a rational factor. Math. Sci. 8(2), Article ID 120 (2014) MathSciNetView ArticleMATHGoogle Scholar
  28. Singh, J, Kumar, D: On the distribution of mixed sum of independent random variables one of them associated with Srivastava’s polynomials and H-function. J. Appl. Math. Stat. Inform. 10, 53-62 (2014) MathSciNetView ArticleMATHGoogle Scholar
  29. Spain, B, Smith, MG: Functions of Mathematical Physics. Van Nostrand Reinhold Company, London (1970). Chapter 10 deals with Laguerre polynomials MATHGoogle Scholar
  30. Spencer, VE: Asymptotic expressions for the zeros of generalized Laguerre polynomials and Weber functions. Duke Math. J. 3(4), 667-675 (1937) MathSciNetView ArticleMATHGoogle Scholar


© Kim et al. 2016