Some identities of Laguerre polynomials arising from differential equations
Kim et al. Advances in Difference Equations
Some identities of Laguerre polynomials arising from differential equations
Taekyun Kim 0 3 4
Dae San Kim 0 2
Kyung-Won Hwang 0 1 6
Jong Jin Seo 0 5
0 Dong-A University , Busan, 49315 , Republic of Korea
1 Department of Mathematics
2 Department of Mathematics, Sogang University , Seoul, 04107 , Republic of Korea
3 Department of Mathematics, Kwangwoon University , Seoul, 139-701 , Republic of Korea
4 Department of Mathematics, College of Science, Tianjin Polytechnic University , Tianjin, 300387 , China
5 Department of Applied mathematics, Pukyong National University , Busan, 48513 , Republic of Korea
6 Department of Mathematics, Dong-A University , Busan, 49315 , Republic of Korea
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
-
1 Introduction
e– x–tt
– t
=
∞
n=
equation
Ln(x)tn
(see [, ]).
xy + ( – x)y + ny =
(see [–]).
From (), we can get the following equation:
∞
n=
Ln(x)tn =
=
=
e– x–tt
– t
=
m=
∞
n= m=
m=
m!
∞ (–)mxmtm ∞
n (–)m n
m x
m
m!
∞ (–)mxmtm
( – t)–m–
m!
l=
m + l
l
tn.
tl
Thus by (), we get immediately the following equation:
Ln(x) =
m=
n (–)m n
m x
m
m!
(n ≥ ) see [, –] .
()
()
()
()
Alternatively, the Laguerre polynomials are also defined by the recurrence relation as
follows:
()
()
L(x) = ,
L(x) = – x,
(n + )Ln+(x) = (n + – x)Ln(x) – nLn–(x) (n ≥ ).
The Rodrigues’ formula for the Laguerre polynomials is given by
Ln(x) = n! ex ddxnn e–xxn
(n ≥ ).
The first few of Ln(x) (n ≥ ) are
L(x) = ,
L(x) = – x,
L(x) = x – x + ,
L(x) = –x + x – x + ,
L(x) = x – x + x – x + .
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 D isotropic
harmonic oscillator (see [, , ]). A contour integral that is commonly taken as the
definition of the Laguerre polynomial is given by
Ln(x) = π i C e –––xttt t–n– dt
see [, , , ] ,
()
where the contour encloses the origin but not the point z = .
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 [–]).
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 [–]).
Many authors have studied the Laguerre polynomials in mathematical physics,
combinatorics and special functions (see [–]). For the applications of special functions and
polynomials, one may referred to the papers (see [, , ]).
F = F(t, x) =
– t
–xt
e –t .
From (), we note that
F() = dFd(tt, x) = ( – t)– – x( – t)– F.
Thus, by (), we get
and
So we are led to put
F() = dF()
F() = dF()
dt
dt
N
i=N
= ( – t)– – x( – t)– + x( – t)– – x( – t)– F.
F(N+) =
ai–N (N , x)i( – t)–i– F +
ai–N (N , x)( – t)–i F()
In [], 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
()
()
()
()
()
()
=
=
=
N
Replacing N by N + in (), we get
F(N+) = i=NN++ ai–N–(N + , x)( – t)–i F.
Comparing the coefficients on both sides of () and (), we have
a(N + , x) = (N + )a(N , x),
and aN+(N + , x) = –xaN (N , x),
ai–N–(N + , x) = iai–N–(N , x) – xai–N–(N , x) (N + ≤ i ≤ N + ).
We note that
F = F() = a(, x)F.
Thus, by (), we get
a(, x) = .
From () and (), we note that
( – t)– – x( – t)– F = F() = a(, x)( – t)– + a(, x)( – t)– F.
Thus, by comparing the coefficients on bot (...truncated)