Laguerre polynomials (named after Edmond Laguerre ) are special polynomials that form an orthogonal system of functions on the interval . They are the solutions of Laguera's differential equation . The Laguerre polynomials play an important role in theoretical physics , especially in quantum mechanics .
![[0, \ infty]](https://wikimedia.org/api/rest_v1/media/math/render/svg/52088d5605716e18068a460dec118214954a68e9)
Differential equation and polynomials
Laguerre's differential equation
The Laguerean differential equation
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,
is an ordinary second order linear differential equation for and![x> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0)
It is a special case of the Sturm-Liouville differential equation
![{\ displaystyle - \ mathrm {e} ^ {x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (x {\ mathrm {e}} ^ {- x} {\ frac {\ mathrm {d} y} {\ mathrm {d} x}} \ right) = ny}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b30e532dc833aa9fbef6b07986227da20a7925b0)
First polynomials
The first five Laguerre polynomials
The first five Laguerre polynomials are
![{\ displaystyle {\ begin {aligned} L_ {0} (x) & = 1 \\ L_ {1} (x) & = - x + 1 \\ L_ {2} (x) & = {\ tfrac {1 } {2}} (x ^ {2} -4x + 2) \\ L_ {3} (x) & = {\ tfrac {1} {6}} (- x ^ {3} + 9x ^ {2} -18x + 6) \\ L_ {4} (x) & = {\ tfrac {1} {24}} (x ^ {4} -16x ^ {3} + 72x ^ {2} -96x + 24) \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a0a2216ee5b3dbf5c69175602e5651627ccaca)
In physics, a definition is usually used according to which the Laguerre polynomials are larger by a factor .
![n!](https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6)
properties
Recursion formulas
The Laguerre polynomial can be combined with the first two polynomials
![{\ displaystyle L_ {n + 1} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec655f990708c830c31e8fd02bb43b18a39bcd00)
![{\ displaystyle L_ {0} (x) = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4731cc01fe5ede29192a6933c9d81ee2a5c544d3)
![{\ displaystyle L_ {1} (x) = 1-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9093330242ab2981aca8223680bb09fe46a262c4)
Calculate using the following recursion formula
![{\ displaystyle (n + 1) L_ {n + 1} (x) = {\ big (} (2n + 1-x) L_ {n} (x) -nL_ {n-1} (x) {\ big )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b32dcf6e1a6ecd3d4a78464527fc00e603006213)
The following recursion formulas also apply:
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,
-
,
-
.
An explicit formula for the Laguerre polynomials is
-
.
- example
The polynomial for is calculated. So
![{\ displaystyle L_ {3} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a71d4ab984051fec77386e3161d79e536f7f1ba)
![n = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34)
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.
In order to obtain this polynomial it is necessary to determine the polynomial for . It turns out
![L_ {2} (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fedf8b2d8f681a5700e030d3e776302ecacf63c)
![n = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425)
![{\ displaystyle L_ {2} (x) = {\ tfrac {1} {2}} {\ big (} (2 + 1-x) L_ {1} (x) -1L_ {0} (x) {\ big)} = {\ tfrac {1} {2}} {\ big (} (3-x) (1-x) -1 {\ big)} = {\ tfrac {1} {2}} (3- 4x + x ^ {2} -1) = {\ tfrac {1} {2}} {\ big (} 2-4x + x ^ {2} {\ big)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23cf107c2118e2d0fff129af9ccb97fae6bf7253)
So the polynomial is
![{\ displaystyle {\ begin {aligned} L_ {3} (x) & = {\ tfrac {1} {3}} {\ big (} (4 + 1-x) {\ tfrac {1} {2}} (2-4x + x ^ {2}) - 2 (1-x) {\ big)} = {\ tfrac {1} {6}} {\ big (} (5-x) (2-4x + x ^ {2}) - 4 + 4x {\ big)} \\ & = {\ tfrac {1} {6}} (10-20x + 5x ^ {2} -2x + 4x ^ {2} -x ^ { 3} -4 + 4x) = {\ tfrac {1} {6}} (6-18x + 9x ^ {2} -x ^ {3}). \ End {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6542fa581cc5eb3b360509ad841cf39ae4d753f)
Rodrigues formula
The -th Laguerre polynomial can be represented with the Rodrigues formula as follows
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle L_ {n} (x) = {\ frac {\ mathrm {e} ^ {x}} {n!}} {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} {\ bigg (} x ^ {n} \ mathrm {e} ^ {- x} {\ bigg)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bdba6869b1ce63bb48d35210864660f26047f3)
and
![{\ displaystyle L_ {n} (x) = {\ frac {1} {n!}} {\ bigg (} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} - 1 {\ bigg)} ^ {n} x ^ {n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1358f5bb5590d5eec6853e7f27a50d498ff06a)
The Laguerre polynomial is calculated from the first equation with the product rule for higher derivatives and the identities , as well as according to
![{\ displaystyle \ textstyle {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} {\ big (} \ mathrm {e} ^ {- x} x ^ { n} {\ big)} = {\ big (} \ mathrm {e} ^ {- x} x ^ {n} {\ big)} ^ {(n)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca3b4113f205da0661cecc0fc4a8f8962552f65)
![{\ displaystyle \ left (\ mathrm {e} ^ {- x} \ right) ^ {(k)} = (- 1) ^ {k} \ mathrm {e} ^ {- x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1066ab323768e44b049e927ffdc731c74d1514ce)
![{\ displaystyle {\ big (} x ^ {n} {\ big)} ^ {(nk)} = {\ tfrac {n!} {k!}} x ^ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6582f9675e98a6b2222959da703fcf50101060cc)
![{\ displaystyle {\ begin {aligned} L_ {n} (x) & = {\ frac {\ mathrm {e} ^ {x}} {n!}} {\ frac {\ mathrm {d} ^ {n} } {\ mathrm {d} x ^ {n}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} = {\ frac {\ mathrm {e} ^ {x}} {n!}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} ^ {(n)} = {\ frac {\ mathrm {e} ^ {x}} {n!}} \ sum _ {k = 0} ^ {n} {\ binom {n} {k}} {\ big (} \ mathrm {e} ^ {- x} {\ big )} ^ {(k)} {\ big (} x ^ {n} {\ big)} ^ {(nk)} \\\\ & = {\ frac {\ mathrm {e} ^ {x}} { n!}} \ sum _ {k = 0} ^ {n} {\ binom {n} {k}} (- 1) ^ {k} \ mathrm {e} ^ {- x} {\ frac {n! } {k!}} x ^ {k} = \ sum _ {k = 0} ^ {n} {\ binom {n} {k}} {\ frac {(-1) ^ {k}} {k! }} x ^ {k}. \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c332261a1af62fa838ae8bbbaca8385238b268e3)
The Laguerre polynomial with the binomial theorem and identity results from the second equation as follows
![{\ displaystyle {\ big (} {\ tfrac {\ mathrm {d}} {\ mathrm {d} x}} {\ big)} ^ {(nk)} x ^ {n} = {\ big (} x ^ {n} {\ big)} ^ {(nk)} = {\ tfrac {n!} {k!}} x ^ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b58af7b8d34b07da0ed1e9f0909dbd01d8b5e05a)
![{\ displaystyle {\ begin {aligned} L_ {n} (x) & = {\ frac {1} {n!}} {\ bigg (} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} - 1 {\ bigg)} ^ {n} x ^ {n} = {\ frac {1} {n!}} {\ bigg (} -1 + {\ frac {\ mathrm {d}} { \ mathrm {d} x}} {\ bigg)} ^ {n} x ^ {n} = {\ frac {1} {n!}} \ sum _ {k = 0} ^ {n} {\ binom { n} {k}} (- 1) ^ {k} {\ bigg (} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ bigg)} ^ {(nk)} x ^ {n} \\\\ & = {\ frac {1} {n!}} \ sum _ {k = 0} ^ {n} {\ binom {n} {k}} (- 1) ^ {k } {\ big (} x ^ {n} {\ big)} ^ {(nk)} = {\ frac {1} {n!}} \ sum _ {k = 0} ^ {n} {\ binom { n} {k}} (- 1) ^ {k} {\ frac {n!} {k!}} x ^ {k} = \ sum _ {k = 0} ^ {n} {\ binom {n} {k}} {\ frac {(-1) ^ {k}} {k!}} x ^ {k}. \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb67e0c20de1cb552fa1c63a919973386814886)
Orthogonal polynomials
Since the Laguerre polynomials are for and / or divergent, they do not form a Prehilbert space or a Hilbert space . Therefore a weight function is introduced which leaves the solution of the differential equation unchanged and which ensures that the Laguerre polynomials can be square-integrated . Under these conditions, the eigenfunctions form an orthonormal basis in the Hilbert space of square-integrable functions with the weight function . Hence applies
![n \ to \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1)
![x \ to \ infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/eda2caf97ec29f30d5f0c0cd7135393361efc020)
![L_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebec334cb04f246db1139e2ca6be0b957d2ef520)
![{\ displaystyle L ^ {2} ([0, \ infty], w (x) \ mathrm {d} x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/619373e228c51e399e63fe05b27c8b1e64e40dfe)
![{\ displaystyle w (x) = \ mathrm {e} ^ {- x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0c3bb74c720c814c166149389377a4d070f9f0)
![{\ displaystyle \ langle L_ {n}, L_ {m} \ rangle = \ int _ {0} ^ {\ infty} \ mathrm {e} ^ {- x} L_ {n} (x) L_ {m} ( x) \ mathrm {d} x = \ delta _ {nm}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa64d401d6bbddf38b8fc411849d8ce271fc864)
Here the Kronecker Delta means .
![\ delta_ {nm}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f596bddc2ec62dcdd068006e870f2ed0324c7384)
- proof
Part 1: First of all, it is shown that the Laguerre polynomials with the weight are orthogonal, so for applies![{\ displaystyle w (x) = \ mathrm {e} ^ {- x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0c3bb74c720c814c166149389377a4d070f9f0)
![n \ neq m](https://wikimedia.org/api/rest_v1/media/math/render/svg/3994a24401e2dfabca26e4f36e53097a07a57af5)
With the Sturm-Liouville operator the following equations result for the Laguerre polynomials :
![{\ displaystyle \ textstyle {\ mathcal {L}} = - \ mathrm {e} ^ {x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (x {\ mathrm { e}} ^ {- x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98200473595b9b9ffda8f93554f2b91197105348)
![{\ displaystyle L_ {n}, L_ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b710ef230f5138f894f95292e03b763c88840fd8)
-
(1)
and
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(2) .
![{\ displaystyle \ quad {\ mathcal {L}} L_ {m} = - \ mathrm {e} ^ {x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (x {\ mathrm {e}} ^ {- x} {\ frac {\ mathrm {d} L_ {m}} {\ mathrm {d} x}} \ right) = mL_ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ad5a06be977d7121a4d61c4a6f24450415766c)
Equation (1) from the left with multiplied by equation (2) , which is also from the left and is multiplied, subtracted, so results in the two equations:
![L_ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205db57bad21096ab2f4d40d5d045fccc2bd07a0)
![L_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebec334cb04f246db1139e2ca6be0b957d2ef520)
-
(3)
and
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(4) .
![{\ displaystyle \ quad L_ {n} {\ mathcal {L}} L_ {m} -L_ {m} {\ mathcal {L}} L_ {n} = (mn) L_ {m} L_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d44b8956a48d279f218155d37a28d9189eadc315)
First, equation (3) is summarized. With the product rule for derivatives , the term is not taken into account, the following presentations result
![{\ displaystyle \ textstyle - \ mathrm {e} ^ {x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f895f0fbd06de79a607f114c5a0c0717fe76df8)
![{\ displaystyle \ textstyle L_ {n} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (x {\ mathrm {e}} ^ {- x} {\ frac {\ mathrm {d} L_ {m}} {\ mathrm {d} x}} \ right) = {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (x {\ mathrm {e} } ^ {- x} L_ {n} {\ frac {\ mathrm {d} L_ {m}} {\ mathrm {d} x}} \ right) - \ left (x {\ mathrm {e}} ^ { -x} {\ frac {\ mathrm {d} L_ {m}} {\ mathrm {d} x}} \ right) {\ frac {\ mathrm {d} L_ {n}} {\ mathrm {d} x }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f496422df663404c100bc9f360e5faaa5a867b68)
and
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.
In this way it can be seen that the second term is the same in both derivatives and disappears when the difference is formed, i.e.:
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(5)
where means the Wronsky determinant of the functions .
![{\ displaystyle W (L_ {n}, L_ {m}) = \ left | {\ begin {smallmatrix} L_ {n} & L_ {m} \\ L_ {n} '& L_ {m}' \ end {smallmatrix} } \ right |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b102d21f2500e4fed770df8757b0980185b4b58e)
![{\ displaystyle L_ {n}, L_ {m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b710ef230f5138f894f95292e03b763c88840fd8)
To calculate the Wronski determinant by means of the Abelian identity is the differential equation or considered, so that a removable singularity in arises. The coefficient matrix of the fundamental system is then and its trace is . Thus the Abelian identity is:
![{\ displaystyle \ textstyle {\ mathcal {L}} y = - \ mathrm {e} ^ {x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (x \ mathrm { e} ^ {- x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ right) y = -xy '' - \ mathrm {e} ^ {x} {\ big (} x \ mathrm {e} ^ {- x} {\ big)} 'y' = - xy '' - {\ big (} 1-x {\ big)} y '= 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/783e5faa63fb45c7d112793dff2d9fb519f7c89c)
![{\ displaystyle \ textstyle y '' + {\ frac {1-x} {x}} y '= 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b0634b386933f5d499e4b4fd5deca5cdbcb6cd)
![x = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc)
![{\ displaystyle \ left ({\ begin {smallmatrix} 0 & 1 \\ 0 & - {\ tfrac {1-x} {x}} \ end {smallmatrix}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c49f2b96f91edfe7b6abf1d8b82efac1d6080ea7)
![{\ displaystyle \ mathrm {Spur} {\ Bigg (} \ left ({\ begin {smallmatrix} 0 & 1 \\ 0 & - {\ tfrac {1-x} {x}} \ end {smallmatrix}} \ right) {\ Bigg)} = - {\ frac {1-x} {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead4487a7b5cba3852347cbd47c10c845b531f6c)
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.
Since and are linearly independent, it is - on closer inspection - and the following result is obtained:
![L_ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebec334cb04f246db1139e2ca6be0b957d2ef520)
![L_ {m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/205db57bad21096ab2f4d40d5d045fccc2bd07a0)
![{\ displaystyle W (L_ {n}, L_ {m}) (0)> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ada42621924aeb13312f065ef39a205cfa985498)
![{\ displaystyle W (L_ {n}, L_ {m}) (0) = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06d7e412d292353d50e0f5366fc68b19fa98591a)
![{\ displaystyle {\ begin {aligned} W (L_ {n}, L_ {m}) (x) & = W (L_ {n}, L_ {m}) (0) \ exp \ left (\ int _ { 0} ^ {x} {\ bigg (} 1 - {\ frac {1} {\ xi}} {\ bigg)} \ mathrm {d} \ xi \ right) = W (L_ {n}, L_ {m }) (0) \ exp {\ Bigg (} {\ bigg [} \ xi - \ ln \ xi {\ bigg]} _ {0} ^ {x} {\ Bigg)} \\ & = \ lim _ { \ xi \ to x} W (L_ {n}, L_ {m}) (0) \ exp {\ Big (} \ xi - \ ln \ xi {\ Big)} - \ lim _ {\ xi \ to 0 } W (L_ {n}, L_ {m}) (0) \ exp {\ Big (} \ xi - \ ln \ xi {\ Big)} \\ & = \ lim _ {\ xi \ to x} W (L_ {n}, L_ {m}) (0) {\ frac {\ exp (\ xi)} {\ exp (\ ln \ xi)}} - \ lim _ {\ xi \ to 0} W (L_ {n}, L_ {m}) (0) {\ frac {\ exp (\ xi)} {\ exp (\ ln \ xi)}} \\ & = \ lim _ {\ xi \ to x} W ( L_ {n}, L_ {m}) (0) {\ frac {\ mathrm {e} ^ {\ xi}} {\ xi}} + \ lim _ {\ xi \ to 0} W (L_ {n} , L_ {m}) (0) {\ frac {\ mathrm {e} ^ {\ xi}} {\ xi}} \\ & = W (L_ {n}, L_ {m}) (0) {\ frac {\ mathrm {e} ^ {x}} {x}} + \ lim _ {\ xi \ to 0} W (L_ {n}, L_ {m}) (0) {\ frac {\ mathrm {e } ^ {\ xi}} {\ xi}} + C. \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/564ae29e9266412c3a404c26fcb0e43f8ed32baf)
The constant of integration is chosen and equation (5) is multiplied by such that it follows:
![{\ displaystyle C = - \ lim _ {\ xi \ to 0} W (L_ {n}, L_ {m}) (0) {\ frac {\ mathrm {e} ^ {\ xi}} {\ xi} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/962664e78eb94d0c1c017a4e0e5e16fe7438f5be)
![{\ displaystyle \ mathrm {e} ^ {- x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b642ff98faa6200cde6e01dc9c88f02c6f5a08fc)
![{\ displaystyle {\ begin {aligned} \ mathrm {e} ^ {- x} {\ big (} L_ {n} {\ mathcal {L}} L_ {m} -L_ {m} {\ mathcal {L} } L_ {n} {\ big)} & = - {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ bigg (} x {\ mathrm {e}} ^ {- x} W (L_ {n}, L_ {m}) (0) {\ frac {\ mathrm {e} ^ {x}} {x}} {\ bigg)} \\ & = - {\ frac {\ mathrm { d}} {\ mathrm {d} x}} {\ bigg (} W (L_ {n}, L_ {m}) (0) {\ bigg)} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a78c3c794389848af9fed5cecfbcbdf2955343a)
After transforming and separating the variables , the equation is now:
![{\ displaystyle - \ mathrm {e} ^ {- x} {\ big (} L_ {n} {\ mathcal {L}} L_ {m} -L_ {m} {\ mathcal {L}} L_ {n} {\ big)} \ mathrm {d} x = \ mathrm {d} {\ bigg (} W (L_ {n}, L_ {m}) (0) {\ bigg)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9070f062761797fbcd83fb28064a12e210c6623)
On both sides of the equation there are now one-dimensional Pfaffian forms and since is a constant function, the following applies . A suitable parameterization must be selected for the calculation of the remaining Pfaffian form . The integral is now:
![{\ displaystyle W (L_ {n}, L_ {m}) (0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69299aaf643bc54344d0ef04eed44a0a4ef09a03)
![{\ displaystyle \ mathrm {d} {\ Big (} W (L_ {n}, L_ {m}) (0) {\ Big)} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd990cb4398dfc65e58e04c0a854a049cca71c22)
![{\ displaystyle \ varphi (t) = t, \ varphi (t_ {0}) = 0, \ varphi (t_ {1}) = \ infty, {\ dot {\ varphi}} (t) = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fb30b9472e2a261e86079593c9f813cc29edcc)
-
.
Accordingly, the integral vanishes along the interval , so that using equation (4) :
![[0, \ infty]](https://wikimedia.org/api/rest_v1/media/math/render/svg/52088d5605716e18068a460dec118214954a68e9)
![{\ displaystyle 0 = (mn) \ int _ {0} ^ {\ infty} \ mathrm {e} ^ {- x} L_ {m} L_ {n} \ mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e60fe505902dec9afc10978de6b5b55864d001f2)
This condition can only be met if:
-
.
Part 2: In the following it is shown that the Laguerre polynomials are bounded with the weight , for is accordingly , or abbreviated .
![n = m](https://wikimedia.org/api/rest_v1/media/math/render/svg/480d6131c6cb07a90f4ec18a376a59fab884b860)
![{\ displaystyle \ langle L_ {n}, L_ {m} \ rangle = \ int \ mathrm {e} ^ {- x} L_ {n} (x) L_ {m} (x) \ mathrm {d} x = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eec2c18385fdaab8b20aa75325556531ad7afe92)
![{\ displaystyle \ langle L_ {n}, L_ {n} \ rangle = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f5c27358e624122aa193bbe6bf93c87ec6cc07e)
The series representation on the one hand and Rodrigues' formula on the other hand are used for the proof . The following applies:
![{\ displaystyle L_ {n} (x) = \ sum _ {k = 0} ^ {n} {\ frac {(-1) ^ {k}} {k!}} x ^ {k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34127aed12d4ff05203a9c82b241edc6e43e2e)
![{\ displaystyle L_ {n} (x) = {\ frac {\ mathrm {e} ^ {x}} {n!}} {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2a84c8743780d09b8ef10ea278a0149e609278)
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For with :
![n = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae)
![{\ displaystyle \ textstyle {\ frac {\ mathrm {d} ^ {n = 0}} {\ mathrm {d} x ^ {n = 0}}} {\ big (} \ mathrm {e} ^ {- x } x ^ {0} {\ big)} = \ mathrm {e} ^ {- x} x ^ {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fcbd864c00590ab9ca684e81643efdbbc0ae2f)
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If the Laguerre polynomial is now decomposed, it follows:
![n> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96)
![{\ displaystyle \ langle L_ {n}, L_ {n} \ rangle = \ sum _ {k = 0} ^ {n-1} {\ frac {(-1) ^ {k}} {k!}} \ int _ {0} ^ {\ infty} x ^ {k} {\ frac {1} {n!}} {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n }}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} \ mathrm {d} x + {\ frac {(-1) ^ {n}} {n! }} \ int _ {0} ^ {\ infty} x ^ {n} {\ frac {1} {n!}} {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} \ mathrm {d} x.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fff008ddd656b19907138b8c5d8396d07a74ade)
This decomposition reduces the total degree of the polynomial by 1 and the following applies , as shown in part 1 . Only the second term remains, which is calculated with partial integration , i.e.:
![{\ displaystyle \ langle L _ {(n-1)}, L_ {n} \ rangle = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6eab150cc5790b7bb776c751d57ccdf3d0f664d)
![{\ displaystyle {\ begin {aligned} \ langle L_ {n}, L_ {n} \ rangle & = {\ frac {(-1) ^ {n}} {n!}} \ int _ {0} ^ { \ infty} x ^ {n} {\ frac {1} {n!}} {\ frac {\ mathrm {d} ^ {n}} {\ mathrm {d} x ^ {n}}} {\ bigg ( } \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} \ mathrm {d} x \\ & = {\ frac {(-1) ^ {n}} {n!}} { \ bigg [} x ^ {n} {\ frac {1} {n!}} {\ frac {\ mathrm {d} ^ {(n-1)}} {\ mathrm {d} x ^ {(n- 1)}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} {\ bigg]} _ {0} ^ {\ infty} -n {\ frac { (-1) ^ {n}} {n!}} \ Int _ {0} ^ {\ infty} x ^ {(n-1)} {\ frac {1} {n!}} {\ Frac {\ mathrm {d} ^ {(n-1)}} {\ mathrm {d} x ^ {(n-1)}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} \ mathrm {d} x \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8905a83663b8b0b8f61eced14ee8409c30ebb852)
The antiderivative was calculated using the product rule and it results in the limit value . The same result is obtained in the limit value . Since this result holds for all partial integrations, it follows:
![{\ displaystyle \ textstyle \ lim _ {x \ to 0} x ^ {n} {\ frac {1} {n!}} {\ frac {\ mathrm {d} ^ {(n-1)}} {\ mathrm {d} x ^ {(n-1)}}} {\ big (} \ mathrm {e} ^ {- x} x ^ {n} {\ big)} = \ sum _ {k = 0} ^ {n-1} \ lim _ {x \ to 0} x ^ {n} {\ frac {1} {n!}} {\ binom {n} {k}} {\ big (} \ mathrm {e} ^ {- x} {\ big)} ^ {k} {\ big (} x ^ {n} {\ big)} ^ {(nk)} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94ee1e3893e982e7bdbe8fe627e1bf19c92c86cd)
![{\ displaystyle \ textstyle \ lim _ {x \ to \ infty}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6e267f999a964746d7062689bb8cc89f5d27c9)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle {\ begin {aligned} \ langle L_ {n}, L_ {n} \ rangle & = (- 1) ^ {1} n {\ frac {(-1) ^ {n}} {n!} } \ int _ {0} ^ {\ infty} x ^ {(n-1)} {\ frac {1} {n!}} {\ frac {\ mathrm {d} ^ {(n-1)}} {\ mathrm {d} x ^ {(n-1)}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} \ mathrm {d} x \\ & = (- 1) ^ {2} n (n-1) {\ frac {(-1) ^ {n}} {n!}} \ Int _ {0} ^ {\ infty} x ^ {(n -2)} {\ frac {1} {n!}} {\ Frac {\ mathrm {d} ^ {(n-2)}} {\ mathrm {d} x ^ {(n-2)}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ {n} {\ bigg)} \ mathrm {d} x \\ & \; \; \ vdots \\ & = (- 1) ^ { n} n! {\ frac {(-1) ^ {n}} {n!}} \ int _ {0} ^ {\ infty} x ^ {(nn)} {\ frac {1} {n!} } {\ frac {\ mathrm {d} ^ {(nn)}} {\ mathrm {d} x ^ {(nn)}}} {\ bigg (} \ mathrm {e} ^ {- x} x ^ { n} {\ bigg)} \ mathrm {d} x \\ & = {\ frac {(-1) ^ {2n}} {n!}} \ int _ {0} ^ {\ infty} \ mathrm {e } ^ {- x} x ^ {n} \ mathrm {d} x \\ & = {\ frac {1} {n!}} \ int _ {0} ^ {\ infty} \ mathrm {e} ^ { -x} x ^ {n} \ mathrm {d} x \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a32796d20050ae17a1f5b2caac140c163394578)
By means of further -fold partial integration or integration table follows and thus:
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle \ textstyle \ int _ {0} ^ {\ infty} \ mathrm {e} ^ {- x} x ^ {n} \ mathrm {d} x = n!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fbb7431bb764b76f00702b4b2360d185ba0cc8)
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Of Part 1 and Part 2 gives:
![{\ displaystyle \ langle L_ {n}, L_ {m} \ rangle = \ int _ {0} ^ {\ infty} \ mathrm {e} ^ {- x} L_ {n} (x) L_ {m} ( x) \ mathrm {d} x = \ delta _ {nm}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa64d401d6bbddf38b8fc411849d8ce271fc864)
![\Box](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Generating function
A generating function for the Laguerre polynomial is
![{\ displaystyle \ sum _ {n = 0} ^ {\ infty} L_ {n} (x) \, t ^ {n} = {\ frac {1} {1-t}} e ^ {- {\ frac {tx} {1-t}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2217b539d1ba0d82a28049bfc975c45d66248e)
Associated Laguerre polynomials
Some associated Laguerre polynomials
The assigned (generalized) Laguerre polynomials overhang with the ordinary Laguerre polynomials
![{\ displaystyle L_ {n} ^ {k} (x) = (- 1) ^ {k} \, {\ frac {{\ rm {d}} ^ {k}} {{\ rm {d}} x ^ {k}}} \, L_ {n + k} (x) \ qquad k = 0.1, \ dotsc}](https://wikimedia.org/api/rest_v1/media/math/render/svg/512df00646271b1428d810928c4628fc80366f04)
together. Your Rodrigues formula is
![L_ {n} ^ {k} (x) = {\ frac {{\ mathrm {e}} ^ {x} \, x ^ {{- k}}} {n!}} \, {\ Frac {{ {\ rm {d}}} ^ {n}} {{{\ rm {d}}} x ^ {n}}} \, ({\ mathrm {e}} ^ {{- x}} \, x ^ {{n + k}}).](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc85cedc3d884476e05fb95e718b4e93b2b7b4d)
The assigned Laguerre polynomials satisfy the assigned Laguerre equation
![{\ displaystyle x \, y '' (x) + (k + 1-x) \, y '(x) + n \, y (x) = 0, \ qquad n = 0.1, \ dotsc}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de7700062018c796a6e28292575e95d488c017f)
The first assigned Laguerre polynomials are:
![L_ {0} ^ {k} (x) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/28cd8f9f53e3eb91a7865548c4e95b682dd93bca)
![L_ {1} ^ {k} (x) = - x + k + 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/31d14cd1b6f6065f0dc1ecef05e320bd78db2e46)
![L_ {2} ^ {k} (x) = {\ frac {1} {2}} \, \ left [x ^ {2} -2 \, (k + 2) \, x + (k + 1) ( k + 2) \ right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae15ad7b67ff44bbd7e3003468ef214b46a99f37)
![{\ displaystyle L_ {3} ^ {k} (x) = {\ frac {1} {6}} \, \ left [-x ^ {3} +3 \, (k + 3) \, x ^ { 2} -3 \, (k + 2) \, (k + 3) \, x + (k + 1) \, (k + 2) \, (k + 3) \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3908ce0eabd5a11f1e577eff87d3829369bfddc5)
The recursion formula
![{\ displaystyle (n + 1) L_ {n + 1} ^ {k} (x) = (2n + 1 + kx) L_ {n} ^ {k} (x) - (n + k) L_ {n- 1} ^ {k} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b263e84aa6162cfe6222da1231bfa5de8a586848)
use.
The Sturm Liouville operator is
![{\ displaystyle {\ mathcal {L}} = - \ mathrm {e} ^ {x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (x ^ {k + 1} \ mathrm {e} ^ {- x} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1813839ed7ca5438a4d9d51cb18f2fe92eaf80)
and with the weight function :
![{\ displaystyle \ mathrm {e} ^ {- x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b642ff98faa6200cde6e01dc9c88f02c6f5a08fc)
![{\ displaystyle \ int \ limits _ {0} ^ {\ infty} \ mathrm {e} ^ {- x} x ^ {k} L_ {m} ^ {k} (x) L_ {n} ^ {k} (x) \ mathrm {d} x = 0 \ qquad m \ neq n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/543e73e7673470d3350cca486ccbeb1ed2a4d13f)
![{\ displaystyle \ int \ limits _ {0} ^ {\ infty} \ mathrm {e} ^ {- x} x ^ {k} \ left (L_ {n} ^ {k} (x) \ right) ^ { 2} \ mathrm {d} x = {\ frac {\ Gamma (n + k + 1)} {n!}} \ Qquad n = 0.1, \ dotsc}](https://wikimedia.org/api/rest_v1/media/math/render/svg/906d66ac1e630d31ce4673b6ef4c2a49489aebf8)
Assigned Laguerre polynomials can be expressed as path integrals :
![{\ displaystyle L_ {n} ^ {k} (x) = {\ frac {1} {2 \ pi i}} \ oint \ limits _ {C} {\ frac {\ mathrm {e} ^ {- {\ frac {xt} {1-t}}}} {(1-t) ^ {k + 1} \, t ^ {n + 1}}} \; dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9693b23e295ecc0e3e6afbd080ea69ad1fc0767f)
There is a path that circles the origin once in a counterclockwise direction and does not include the essential singularity at 1.
![C.](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029)
Hydrogen atom
The Laguerre polynomials are used in quantum mechanics to solve the Schrödinger equation for the hydrogen atom or, in the general case, for a Coulomb potential. Using the assigned Laguerre polynomials, the radial component of the wave function can be written as
![R _ {{nl}} (r) = D _ {{nl}} \, {\ mathrm {e}} ^ {{- \ kappa \, r}} \, (2 \, \ kappa \, r) ^ { l} \, L _ {{nl-1}} ^ {{2 \, l + 1}} (2 \, \ kappa \, r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae5e9d084542ddcaf17a8f28d891bca03debe17)
(Normalization constant , characteristic length , principal quantum number, orbital angular momentum quantum number ). The assigned Laguerre polynomials therefore play a decisive role here. The normalized total wave function is through
![D _ {{nl}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a674d681aa496e957575365a4eb51dc85123d7)
![\ kappa](https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![l](https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac)
![{\ displaystyle \ Psi _ {n, l, m} (r, \ vartheta, \ varphi) = {\ sqrt {\ frac {4 \, (nl-1)!} {(n + l)! \; n \, (na_ {0} / Z) ^ {3}}}} \ left [{\ frac {2r} {na_ {0} / Z}} \ right] ^ {l} \ exp {\ left \ {- {\ frac {r} {na_ {0} / Z}} \ right \}} \; L_ {nl-1} ^ {2l + 1} \ left ({\ frac {2r} {na_ {0} / Z }} \ right) \; Y_ {l, m} (\ vartheta, \ varphi)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45c656de94c4f217677df1ae610d165293d61938)
optionally, with the principal quantum number , the orbital angular momentum quantum number , the magnetic quantum number , the Bohr radius and the atomic number . The functions
are the assigned Laguerre polynomials,
the spherical surface functions .
![Z](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd)
![L_ {n} ^ {l} (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5037d37c446dc591aaeaffbda742d56694ed3e15)
![Y _ {{l, m}} (\ vartheta, \ varphi)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee601e62373db1126c962d12cff3971bfa907ef)
Web links
References and comments
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↑ Because of the linear parameterization, the differential can be selected as it is.
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↑ In physics, the term standardized is usually used instead of restricted.
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↑ Harro Heuser : Ordinary differential equations , Vieweg + Teubner 2009 (6th edition), pages 352–354, ISBN 978-3-8348-0705-2