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“laguerre-polynomials1a” — 2017/10/2 — 14:06 — page 3 — #1
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APORTAC IO NE S MATE MÁTICAS
Memorias 52 (2017) 3–13
Artículo de Investigación
LAGUERRE POLYNOMIALS AND LINEAR ALGEBRA
VICENTE ABOITES
ABSTRACT . Laguerre Polynomials are obtained in a straightforward and sim-
ple way, throughwell known linear algebra methods based on Sturm-Liouville
theory. A matrix corresponding to the Laguerre differential operator is found
and its eigenvalues are obtained. The elements of the eigenvectors obtained,
corresponds to the Laguerre polynomials. This simplicity contrast with the
complexity normally found when solving the Laguerre equation by power se-
ries, obtaining Laguerre polynomials through a generating function or using
the Rodriguez formula, or through a contour integral.
RESUMEN. Usando métodos estándar de álgebra lineal basados en la teoría
de Sturm-Liouville, se obtienen los polinomios de Laguerre de un modo sim-
ple y directo. Se determina la matriz asociada a la transformación lineal que
define el operador de Laguerre en el espacio de polinomios y se hacen ex-
plícitos los cálculos de eigenvalores y eigenvectores, estos últimos se cor-
responden con los polinomios de Laguerre. Esta simplicidad contrasta con
la complejidad normalmente encontrada cuando se resuelve la ecuación de
Laguerre por series de potencias, o cuando se obtienen los polinomios de
Laguerre a partir de una función generadora o de la fórmula de Rodríguez, o
a través de una integral de contorno.
2010 Mathematics Subject Classification. 33B99, 42C05, 34L15 .
Keywords and phrases. Laguerre, Laguerre polynomials, special functions.
3
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“laguerre-polynomials1a” — 2017/10/2 — 14:06 — page 4 — #2
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4 VICENTE ABOITES
1. INTRODUCTION.
Laguerre differential equation and its solutions, i.e. Laguerre polynomials, are found
in many important physical problems. The study of the hydrogen atom is an impor-
tant example [1], another one is the solution of the paraxial wave equation in cylindri-
cal coordinates in optics and laser physics [2]. Many more applications are found in
systems theory and coding theory. Laguerre polynomials are studied in most science
and engineering mathematics courses, mainly in those courses focused on differential
equations and special functions.
These polynomials are typically obtained as a result of the solution of Laguerre dif-
ferential equation by power series. Usually it is also shown that they can be obtained
by a generating function and also by Rodriguez formula for Laguerre polynomial or
through a contour integral. Most courses also include a study of the properties of these
polynomials such as: orthogonality, completeness, recursion relations, special values,
asymptotic expansions and relation to other functions such as polynomials and hyper-
geometric functions [3], [4]. There is no doubt that this is a challenging and demanding
subject that requires a great deal of attention from most students. The Sturm-Liouville
Theory is covered in most advanced courses. In this context an eigenvalue equation
sometimes takes the more general self-adjoint form: Lu(x)+λw(x)u(x)=0, where L
is a differential operator; Lu(x)=d
d x hp(x)du(x)
d x i+q(x)u(x), λan eigenvalue, and w(x)
is known as a weight or density function. The analysis of this equation and its solutions
is called Sturm-Liouville theory. Specific forms of p(x), q(x), λand w(x), are given for
Legendre, Laguerre, Hermite and other well-known equations in the given references.
There, it is also shown the close analogy of this theory with linear algebra concepts.
For example, function here take the role of vectors there, and linear operators here
take that of matrices there. Finally, the diagonalization of a real symmetric matrix cor-
responds to the solution of an ordinary differential equation, defined by a self-adjoint
operator L, in terms of its eigenfunctions which are the "continuous" analog of the
eigenvectors.
In this paper Laguerre polynomials are obtained using basic concepts of linear al-
gebra (which most students are already familiar with) and which contrasts in simplic-
ity with the standard methods as those described in the previously outlined syllabus.
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LAGUERRE POLYNOMIALS AND LINEAR ALGEBRA 5
In the next section the Laguerre differential operator matrix is obtained as well as its
eigenvalues and eigenvectors. From the eigenvectors found, the Laguerre polynomials
follow.
2. LAGUERRE POLYNOM IA LS
Laguerre polynomials are solutions of Laguerre equation:
(1) xd2y
d x2+(1 −x)dy
d x +λy=0 .
The associated Laguerre differential equation has a regular singular point at 0 and
an irregular singularity at ∞. In most mathematics courses this equation is solved us-
ing a series expansion:
(2) y=
∞
X
n=0
anxn|.
Substituting the proposed solution in the equation one obtains:
(3) x
∞
X
n=0
n(n−1)anxn−2+
∞
X
n=1
nanxn−1−x
∞
X
n=1
nanxn−1+λ
∞
X
n=0
anxn=0
(4)
∞
X
n=2
n(n−1)anxn−1+
∞
X
n=1
nanxn−1−
∞
X
n=1
nanxn+λ
∞
X
n=0
anxn=0
(5)
∞
X
n=1
n(n+1)an+1xn+
∞
X
n=0
(n+1)an+1xn−
∞
X
n=1
nanxn+λ
∞
X
n=0
anxn=0
(6) [a1+λa0]+
∞
X
n=0
{[(n+1)n+(n+1)]an+1−nan+λan}xn=0
(7) [a1+λa0]+
∞
X
n=1
[(n+1)(n+1)an+1+(λ−n)an]xn=0
This requires
(8) a1= −λa0
(9) an+1=n−λ
(n+1)(n+1) an
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6 VICENTE ABOITES
For n=1, 2, ·· ·
(10) y=
∞
X
n=0
anxn=a0·1−λx−
λ(1−λ)
2·2x2−
λ(1−λ)(2 −λ)
2·3·2·3x3+ · · · ¸
If λis a nonnegative integer, the series terminate and the solution is given by La-
guerre polynomial
(11) y=a0Lλ(x)
The same solution may also be obtained using other methods such as the Rodriguez
formula, the generating function and a contour integral.
The Rodriguez representation of Laguerre polynomials is:
(12) Lλ(x)=ex
n!
dn
d xn(xne−x)=1
n!µd
d x −1¶n
xn
The generating function for Laguerre polynomials is:
(13) g(x,z)=exp ¡−x z
1−z¢
1−z=
∞
X
n=0
Ln(x)zn
or:
(14) g(x,z)=1+(−x+1)z+µ1
2x2−2x+1¶z2+µ−1
6x3+3
2x2−3x+1¶z3+ · · ·
Laguerre polynomials may also be defined as a contour integral given by:
(15) Ln(x)=1
2πiIe−xz /(1−z)
(1−z)zn+1d z
Next, Laguerre polynomials will be obtained using simple linear algebra concepts.
The algebraic polynomial of degree n,
(16) a0+a1x+a2x2+a3x3+ · · · + anxn
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LAGUERRE POLYNOMIALS AND LINEAR ALGEBRA 7
with a0,a1, ..., an∈R, is represented by the vector:
(17) An=
a0
a1
a2
a3
.
.
.
an
Taking first derivative of the above polynomial (16) one obtains the polynomial:
(18) d
d x ¡a0+a1x+a2x2+a3x3+ · · · + anxn¢=a1+2a2x+3a3x2+· · · +n anxn−1,
Which may be written as:
(19) d An
d x =
a1
2a2
3a3
.
.
.
nan
0
Taking the second derivative of polynomial (16) one obtains:
(20) d2
d x2¡a0+a1x+a2x2+a3x3+ ·· · + anxn¢=
2a2+6a3x+12a4x2+ · · · + n(n−1)anxn−2,
or,
(21) d2An
d x2=
2a2
6a3
.
.
.
n(n−1)an
0
0
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8 VICENTE ABOITES
Equation (19) may be written as:
(22)
0 1 0 0 · · · 0
0 0 2 0 · · · 0
0 0 0 3 · · · 0
.
.
..
.
..
.
..
.
.....
.
.
0 0 0 0 · · · n
0 0 0 0 · · · 0
a0
a1
a2
a3
.
.
.
an
=
a1
2a2
3a3
.
.
.
nan
0
Therefore the first derivative operator of Anmay be written as:
(23) d
d x →
0 1 0 0 · · · 0
0 0 2 0 · · · 0
0 0 0 3 · · · 0
.
.
..
.
..
.
..
.
.....
.
.
0 0 0 0 · · · n
0 0 0 0 · · · 0
In a similar manner, equation (21) may be written as:
(24)
0 0 2 0 · · · 0
0 0 0 6 · · · 0
.
.
..
.
..
.
..
.
.....
.
.
0 0 0 0 · · · n(n−1)
0 0 0 0 · · · 0
0 0 0 0 · · · 0
a0
a1
a2
a3
.
.
.
an
=
2a2
6a3
.
.
.
n(n−1)an
0
0
Therefore the second derivative operator of Anmay be written as:
(25) d2
d x2→
0 0 2 0 · · · 0
0 0 0 6 · · · 0
.
.
..
.
..
.
..
.
.....
.
.
0 0 0 0 · · · n(n−1)
0 0 0 0 · · · 0
0 0 0 0 · · · 0
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LAGUERRE POLYNOMIALS AND LINEAR ALGEBRA 9
The Laguerre differential operator is given by:
(26) xd2
d x2+(1 −x)d
d x
Which using eq. (8) and (10) may be written as:
(27) x[2a2+6a3x+12a4x2+20a5x4+ · · · + n(n−1)anxn−2]+
(1−x)[a1+2a2x+3a3x2+4a4x3+ · · · + nanxn−1]
(28) =[2a2x+6a3x2+12a4x3+20a5x4+ · · · + n(n−1)anxn−1]+
[a1+2a2x+3a3x2+4a4x3+ · · · + n anxn−1]+
−[a1x+2a2x2+3a3x3+4a4x4+ · · · + n anxn]
(29) =a1+(4a2−a1)x+(9a3−2a2)x2+(16a4−3a3)x3+ · · · − n anxn
Which may be written as:
(30)
0 1 0 0 0 · · · 0
0−1 4 0 0 · · · 0
0 0 −2 9 0 · · · 0
0 0 0 −3 16 · · · 0
.
.
..
.
..
.
..
.
..
.
.....
.
.
0 0 0 0 0 · · · −n
a0
a1
a2
a3
.
.
.
an−1
an
=
a1
4a2−a1
9a3−2a2
16a4−3a3
.
.
.
nan
Therefore as an example and for the sake of simplicity, the Laguerre differential op-
erator as a 5×5 matrix is represented by the following matrix:
(31) xd2
d x2+(1 −x)d
d x →
01000
0−1 4 0 0
0 0 −2 9 0
0 0 0 −3 16
0 0 0 0 −4
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10 VICENTE ABOITES
The eigenvalues of a matriz M are the values that satisfy the equation De t (M−λI)=
0. However since the matrix (31) is a triangular matrix, the eigenvalues λiof this matrix
are the elements of the diagonal, namely: λ1=0, λ2= −1, λ3= −2, λ4= −3, λ5=
−4. The corresponding eigenvectors are the solutions of the equation (M−λiI)·v=0,
where the eigenvector v=[a0,a1,a2,a3,a4]T.
(32)
0−λi1 0 0 0
0−1−λi4 0 0
0 0 −2−λi9 0
0 0 0 −3−λi16
0 0 0 0 −4−λi
a0
a1
a2
a3
a4
=
0
0
0
0
0
Substituting in equation (16) the first eigenvalue λ1= 0, one obtains the eigenvector
equation for v1:
(33)
01000
0−1 4 0 0
0 0 −2 9 0
0 0 0 −3 16
0 0 0 0 −4
a0
a1
a2
a3
a4
=
0
0
0
0
0
The eigenvector v1is:
(34)
a0
a1
a2
a3
a4
=
1
0
0
0
0
The elements of this eigenvector corresponds to the first Laguerre polynomial,
L0(x)=1.
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LAGUERRE POLYNOMIALS AND LINEAR ALGEBRA 11
Substituting in equation (32) the second eigenvalue λ2= −1, one obtains the eigen-
vector equation for v2:
(35)
1 1 0 0 0
0 0 4 0 0
0 0 −1 9 0
0 0 0 −2 16
0 0 0 0 −3
a0
a1
a2
a3
a4
=
0
0
0
0
0
The eigenvector v2is:
(36)
a0
a1
a2
a3
a4
=
1
−1
0
0
0
The elements of this eigenvector corresponds to the second Laguerre polynomial,
L1(x)=1−x.
Substituting in equation (32) the third eigenvalue λ3= −2, one obtains the eigen-
vector equation for v3:
(37)
2 1 0 0 0
0 1 4 0 0
0 0 0 9 0
0 0 0 −1 16
0 0 0 0 −2
a0
a1
a2
a3
a4
=
0
0
0
0
0
The eigenvector v3is:
(38)
a0
a1
a2
a3
a4
=
1
−2
1/2
0
0
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12 VICENTE ABOITES
The elements of this eigenvector corresponds to the third Laguerre polynomial,
L3(x)=1−2x+x2/2.
Substituting in equation (32) the fourth eigenvalue λ4= −3, one obtains the eigen-
vector equation for v4:
(39)
3 1 0 0 0
0 2 4 0 0
0 0 1 9 0
0 0 0 0 16
0 0 0 0 −1
a0
a1
a2
a3
a4
=
0
0
0
0
0
The eigenvector v4is:
(40)
a0
a1
a2
a3
a4
=
1
−3
3/2
−1/6
0
The elements of this eigenvector corresponds to the fourth Laguerre polynomial,
L3(x)=1−3x+3x2/2−x3/6.
Using a larger matrix, higher order polynomials may be found, however the gen-
eral case for a nx n matrix was not obtained since it seems that, in this general case,
standard methods would be easier. The main advantage of this method stands in its
easiness because it relies on simple linear algebra concepts. Following this method
Hermite polynomials were also obtained [5], as well as Legendre and Tchebicheff. This
pedagogical material has been successfully used by the author with first year university
students.
3. CONCLUSION
Laguerre polynomials are obtained using basic linear algebra concepts such the eigen-
value and eigenvector of a matrix. Once the corresponding matrix of the Laguerre
differential operator is obtained, the eigenvalues of this matrix are found and the el-
ements of its eigenvectors correspond to the coefficients of Laguerre Polynomials.
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LAGUERRE POLYNOMIALS AND LINEAR ALGEBRA 13
REFERENCES
[1] M. Abramowitz, I. A. Stegun, ‘Handbook of Mathematical Functions, With Formulas,
Graphs, and Mathematical Tables’, Dover Publications (2012).
[2] P. Frank, R. v. Mises, Die Differential und Integralgleichungen der Mechanik und Physik,
Ed. Mary S. Rosenberg, New York, (1943)
[3] M.L. Boas, ‘Mathematical Methods in the Physical Sciences’, John Wiley & Sons, (2014).
[4] H. J. Weber, G.B. Arfken, ‘Mathematical Methods for Physicists’, Academic Press, (2011).
[5] V. Aboites, Hermite polynomials through linear algebra, International Journal of Pure and
Applied Mathematics, 114 (2), (2017), 401–406
LABORATORIO DE LÁSERES, CENTRO D E INV ES TI GACIÓN EN ÓPTICA, LOMA DEL BOSQUE 115,
COL. CAMPESTRE, 37150 LEÓN, MÉXICO
E-mail address:aboites@cio.mx
