→Physicist Scaling Convention: Section read like written by non-native English speaker. The meat of the section didn't change, and if anything, this section should go higher in the article.
The generalized Laguerre polynomials are used to describe the quantum wavefunction for [[hydrogen atom]] orbitals.<ref>{{cite book |last1=Griffiths |first1=David J. |title=Introduction to quantum mechanics |date=2005 |publisher=Pearson Prentice Hall |location=Upper Saddle River, NJ |isbn=0131118927 |edition=2nd}}</ref><ref>{{cite book |last1=Sakurai |first1=J. J. |title=Modern quantum mechanics |date=2011 |publisher=Addison-Wesley |location=Boston |isbn=978-0805382914 |edition=2nd}}</ref><ref name="Merzbacher">{{cite book |last1=Merzbacher |first1=Eugen |title=Quantum mechanics |date=1998 |publisher=Wiley |location=New York |isbn=0471887021 |edition=3rd}}</ref> The convention used throughout this article expressesthe generalized Laguerre polynomials as <ref>{{cite book |last1=Abramowitz |first1=Milton |title=Handbook of mathematical functions, with formulas, graphs, and mathematical tables |date=1965 |publisher=Dover Publications |location=New York |isbn=978-0-486-61272-0}}</ref>
The generalized Laguerre polynomials are used to describe the quantum wavefunction for [[hydrogen atom]] orbitals.<ref>{{cite book |last1=Griffiths |first1=David J. |title=Introduction to quantum mechanics |date=2005 |publisher=Pearson Prentice Hall |location=Upper Saddle River, NJ |isbn=0131118927 |edition=2nd}}</ref><ref>{{cite book |last1=Sakurai |first1=J. J. |title=Modern quantum mechanics |date=2011 |publisher=Addison-Wesley |location=Boston |isbn=978-0805382914 |edition=2nd}}</ref><ref name="Merzbacher">{{cite book |last1=Merzbacher |first1=Eugen |title=Quantum mechanics |date=1998 |publisher=Wiley |location=New York |isbn=0471887021 |edition=3rd}}</ref> The convention used throughout this article expresses the generalized Laguerre polynomials as <ref>{{cite book |last1=Abramowitz |first1=Milton |title=Handbook of mathematical functions, with formulas, graphs, and mathematical tables |date=1965 |publisher=Dover Publications |location=New York |isbn=978-0-486-61272-0}}</ref>
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
The first six Laguerre polynomials.
Recursive definition, closed form, and generating function
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
and then using the following recurrence relation for any k ≥ 1:
Furthermore,
In solution of some boundary value problems, the characteristic values can be useful:
Polynomials of negative index can be expressed using the ones with positive index:
Generalized Laguerre polynomials
For arbitrary real α the polynomial solutions of the differential equation[2]
are called generalized Laguerre polynomials, or associated Laguerre polynomials.
One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as
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 polynomials' asymptotic behaviour for large n, but fixed α and x > 0, is given by[6][7] and summarizing by where is the Bessel function.
As a contour integral
Given the generating function specified above, the polynomials may be expressed in terms of a contour integral
where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1
Laguerre's polynomials satisfy the recurrence relations
in particular
and
or
moreover
They can be used to derive the four 3-point-rules
combined they give this additional, useful recurrence relations
Since is a monic polynomial of degree in ,
there is the partial fraction decomposition
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.
Moreover, the following equation holds:
which generalizes with Cauchy's formula to
The derivative with respect to the second variable α has the form,[9]
This is evident from the contour integral representation below.
The generalized Laguerre polynomials obey the differential equation
which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,
This leads directly to
for the exponential function. The incomplete gamma function has the representation
In quantum mechanics
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]
The generalized Laguerre polynomials are related to the Hermite polynomials:
where the Hn(x) are the Hermite polynomials based on the weighting function exp(−x2), the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.
The generalized Laguerre polynomials satisfy the Hardy–Hille formula[14][15]
where the series on the left converges for and . Using the identity
(see generalized hypergeometric function), this can also be written as
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 bewhere 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.