Sometimes the name Laguerre polynomials is used for solutions of
where n is still a non-negative integer.
Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor[1]Nikolay Yakovlevich Sonin).
More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer.
The Laguerre polynomials are also used for
Gauss–Laguerre quadrature to numerically compute integrals of the form
The
rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the
Tricomi–Carlitz polynomials.
Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
The first few polynomials
These are the first few Laguerre polynomials:
n
0
1
2
3
4
5
6
n
Recursive definition, closed form, and generating function
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
where is a generalized
binomial coefficient. When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression[4]
Laguerre polynomials have a differential operator representation, much like the closely related Hermite polynomials. Namely, let and consider the differential operator . Then .[citation needed]
The first few generalized Laguerre polynomials are:
The second equality follows by the following identity, valid for integer i and n and immediate from the expression of in terms of
Charlier polynomials:
For the third equality apply the fourth and fifth identities of this section.
Derivatives of generalized Laguerre polynomials
Differentiating the
power series representation of a generalized Laguerre polynomial k times leads to
This points to a special case (α = 0) of the formula above: for integer α = k the generalized polynomial may be written
the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.
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.[11]
Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.[12]
This formula is a generalization of the
Mehler kernel for
Hermite polynomials, which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above.
Physics Convention
The generalized Laguerre polynomials are used to describe the quantum wavefunction for
hydrogen atom orbitals.[16][17][18] The convention used throughout this article expresses the generalized Laguerre polynomials as [19]
The physics version is related to the standard version by
There is yet another, albeit less frequently used, convention in the physics literature [20][21][22]
Umbral Calculus Convention
Generalized Laguerre polynomials are linked to
Umbral calculus by being
Sheffer sequences for when multiplied by . In Umbral Calculus convention,[23] the default Laguerre polynomials are defined to be
where are the signless
Lah numbers. is a sequence of polynomials of
binomial type, ie they satisfy
^D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", SIAM J. Numer. Anal., vol. 46 (2008), no. 6, pp. 3285–3312
doi:
10.1137/07068031X
^Koepf, Wolfram (1997). "Identities for families of orthogonal polynomials and special functions". Integral Transforms and Special Functions. 5 (1–2): 69–102.
CiteSeerX10.1.1.298.7657.
doi:
10.1080/10652469708819127.
^Ratner, Schatz, Mark A., George C. (2001). Quantum Mechanics in Chemistry. 0-13-895491-7: Prentice Hall. pp. 90–91.{{
cite book}}: CS1 maint: location (
link) CS1 maint: multiple names: authors list (
link)
^Griffiths, David J. (2005). Introduction to quantum mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall.
ISBN0131118927.
^Sakurai, J. J. (2011). Modern quantum mechanics (2nd ed.). Boston: Addison-Wesley.
ISBN978-0805382914.
^
abMerzbacher, Eugen (1998). Quantum mechanics (3rd ed.). New York: Wiley.
ISBN0471887021.
^Abramowitz, Milton (1965). Handbook of mathematical functions, with formulas, graphs, and mathematical tables. New York: Dover Publications.
ISBN978-0-486-61272-0.
^Schiff, Leonard I. (1968). Quantum mechanics (3d ed.). New York: McGraw-Hill.
ISBN0070856435.
^Messiah, Albert (2014). Quantum Mechanics. Dover Publications.
ISBN9780486784557.
^Boas, Mary L. (2006). Mathematical methods in the physical sciences (3rd ed.). Hoboken, NJ: Wiley.
ISBN9780471198260.
Sometimes the name Laguerre polynomials is used for solutions of
where n is still a non-negative integer.
Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor[1]Nikolay Yakovlevich Sonin).
More generally, a Laguerre function is a solution when n is not necessarily a non-negative integer.
The Laguerre polynomials are also used for
Gauss–Laguerre quadrature to numerically compute integrals of the form
The
rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the
Tricomi–Carlitz polynomials.
Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
The first few polynomials
These are the first few Laguerre polynomials:
n
0
1
2
3
4
5
6
n
Recursive definition, closed form, and generating function
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
where is a generalized
binomial coefficient. When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression[4]
Laguerre polynomials have a differential operator representation, much like the closely related Hermite polynomials. Namely, let and consider the differential operator . Then .[citation needed]
The first few generalized Laguerre polynomials are:
The second equality follows by the following identity, valid for integer i and n and immediate from the expression of in terms of
Charlier polynomials:
For the third equality apply the fourth and fifth identities of this section.
Derivatives of generalized Laguerre polynomials
Differentiating the
power series representation of a generalized Laguerre polynomial k times leads to
This points to a special case (α = 0) of the formula above: for integer α = k the generalized polynomial may be written
the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.
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.[11]
Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.[12]
This formula is a generalization of the
Mehler kernel for
Hermite polynomials, which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above.
Physics Convention
The generalized Laguerre polynomials are used to describe the quantum wavefunction for
hydrogen atom orbitals.[16][17][18] The convention used throughout this article expresses the generalized Laguerre polynomials as [19]
The physics version is related to the standard version by
There is yet another, albeit less frequently used, convention in the physics literature [20][21][22]
Umbral Calculus Convention
Generalized Laguerre polynomials are linked to
Umbral calculus by being
Sheffer sequences for when multiplied by . In Umbral Calculus convention,[23] the default Laguerre polynomials are defined to be
where are the signless
Lah numbers. is a sequence of polynomials of
binomial type, ie they satisfy
^D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", SIAM J. Numer. Anal., vol. 46 (2008), no. 6, pp. 3285–3312
doi:
10.1137/07068031X
^Koepf, Wolfram (1997). "Identities for families of orthogonal polynomials and special functions". Integral Transforms and Special Functions. 5 (1–2): 69–102.
CiteSeerX10.1.1.298.7657.
doi:
10.1080/10652469708819127.
^Ratner, Schatz, Mark A., George C. (2001). Quantum Mechanics in Chemistry. 0-13-895491-7: Prentice Hall. pp. 90–91.{{
cite book}}: CS1 maint: location (
link) CS1 maint: multiple names: authors list (
link)
^Griffiths, David J. (2005). Introduction to quantum mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall.
ISBN0131118927.
^Sakurai, J. J. (2011). Modern quantum mechanics (2nd ed.). Boston: Addison-Wesley.
ISBN978-0805382914.
^
abMerzbacher, Eugen (1998). Quantum mechanics (3rd ed.). New York: Wiley.
ISBN0471887021.
^Abramowitz, Milton (1965). Handbook of mathematical functions, with formulas, graphs, and mathematical tables. New York: Dover Publications.
ISBN978-0-486-61272-0.
^Schiff, Leonard I. (1968). Quantum mechanics (3d ed.). New York: McGraw-Hill.
ISBN0070856435.
^Messiah, Albert (2014). Quantum Mechanics. Dover Publications.
ISBN9780486784557.
^Boas, Mary L. (2006). Mathematical methods in the physical sciences (3rd ed.). Hoboken, NJ: Wiley.
ISBN9780471198260.