LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle L_{-n}(x)=e^xL_{n-1}(-x).}

${\displaystyle L_{-n}(x)=e^{x}L_{n-1}(-x).}$

... is translated to the CAS output ...

Semantic latex: L_{-n}(x) = \expe^x L_{n-1}(- x)

Confidence: 0

Mathematica

Translation: Subscript[L, - n][x] == Exp[x]*Subscript[L, n - 1][- x]

Information

Sub Equations

Subscript[L, - n][x] = Exp[x]*Subscript[L, n - 1][- x]

Free variables

n

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Recognizes e with power as the exponential function. It was translated as a function.

Tests

Symbolic

Numeric

SymPy

Translation: Symbol('{L}_{- n}')(x) == exp(x)*Symbol('{L}_{n - 1}')(- x)

Information

Sub Equations

Symbol('{L}_{- n}')(x) = exp(x)*Symbol('{L}_{n - 1}')(- x)

Free variables

n

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Recognizes e with power as the exponential function. It was translated as a function.

Tests

Symbolic

Numeric

Maple

Translation: L[- n](x) = exp(x)*L[n - 1](- x)

Information

Sub Equations

L[- n](x) = exp(x)*L[n - 1](- x)

Free variables

n

x

Symbol info

Function without DLMF-Definition. We keep it like it is (but delete prefix \ if necessary).

Recognizes e with power as the exponential function. It was translated as a function.

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$L_{-n}(x)=e^{x}L_{n-1}(-x)$

$n$

Is part of

$L_{-n}(x)=e^{x}L_{n-1}(-x)$

Complete translation information:

{
  "id" : "FORMULA_42a0b0241aac93e77b485fdad16e56d1",
  "formula" : "L_{-n}(x)=e^x L_{n-1}(-x)",
  "semanticFormula" : "L_{-n}(x) = \\expe^x L_{n-1}(- x)",
  "confidence" : 0.0,
  "translations" : {
    "Mathematica" : {
      "translation" : "Subscript[L, - n][x] == Exp[x]*Subscript[L, n - 1][- x]",
      "translationInformation" : {
        "subEquations" : [ "Subscript[L, - n][x] = Exp[x]*Subscript[L, n - 1][- x]" ],
        "freeVariables" : [ "n", "x" ],
        "tokenTranslations" : {
          "L" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "SymPy" : {
      "translation" : "Symbol('{L}_{- n}')(x) == exp(x)*Symbol('{L}_{n - 1}')(- x)",
      "translationInformation" : {
        "subEquations" : [ "Symbol('{L}_{- n}')(x) = exp(x)*Symbol('{L}_{n - 1}')(- x)" ],
        "freeVariables" : [ "n", "x" ],
        "tokenTranslations" : {
          "L" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    },
    "Maple" : {
      "translation" : "L[- n](x) = exp(x)*L[n - 1](- x)",
      "translationInformation" : {
        "subEquations" : [ "L[- n](x) = exp(x)*L[n - 1](- x)" ],
        "freeVariables" : [ "n", "x" ],
        "tokenTranslations" : {
          "L" : "Function without DLMF-Definition. We keep it like it is (but delete prefix \\ if necessary).",
          "\\expe" : "Recognizes e with power as the exponential function. It was translated as a function."
        }
      },
      "numericResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "wasAborted" : false,
        "crashed" : false,
        "testCalculationsGroups" : [ ]
      },
      "symbolicResults" : {
        "overallResult" : "SKIPPED",
        "numberOfTests" : 0,
        "numberOfFailedTests" : 0,
        "numberOfSuccessfulTests" : 0,
        "numberOfSkippedTests" : 0,
        "numberOfErrorTests" : 0,
        "crashed" : false,
        "testCalculationsGroup" : [ ]
      }
    }
  },
  "positions" : [ ],
  "includes" : [ "L_{-n}(x)=e^xL_{n-1}(-x)", "n" ],
  "isPartOf" : [ "L_{-n}(x)=e^xL_{n-1}(-x)" ],
  "definiens" : [ ]
}

Specify your own input

Formula input
Wikitext context	In [[mathematics]], the '''Laguerre polynomials''', named after [[Edmond Laguerre]] (1834–1886), are solutions of '''Laguerre's equation:''' : <math>xy'' + (1 - x)y' + ny = 0</math> which is a second-order [[linear differential equation]]. This equation has nonsingular solutions only if ''n'' is a non-negative integer. Sometimes the name '''Laguerre polynomials''' is used for solutions of : <math>xy'' + (\alpha+1 - x)y' + ny = 0~.</math> where {{mvar\|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<ref>{{cite journal\|title=Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries\|author=N. Sonine\|journal=[[Math. Ann.]]\|date=1880\|volume=16\|issue=1\|pages=1–80\|doi=10.1007/BF01459227\|url=http://www.digizeitschriften.de/dms/img/?PPN=PPN235181684_0016&DMDID=dmdlog8\|author-link=N. Sonine}}</ref> [[Nikolay Yakovlevich Sonin]]). More generally, a '''Laguerre function''' is a solution when {{mvar\|n}} is not necessarily a non-negative integer. The Laguerre polynomials are also used for [[Gaussian quadrature]] to numerically compute integrals of the form : <math>\int_0^\infty f(x) e^{-x} \, dx.</math> These polynomials, usually denoted ''L''<sub>0</sub>, ''L''<sub>1</sub>, ..., are a [[polynomial sequence]] which may be defined by the [[Rodrigues formula#Rodrigues formula\|Rodrigues formula]], :<math>L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right) =\frac{1}{n!} \left( \frac{d}{dx} -1 \right)^n x^n,</math> reducing to the closed form of a following section. They are [[orthogonal polynomials]] with respect to an [[inner product]] : <math>\langle f,g \rangle = \int_0^\infty f(x) g(x) e^{-x}\,dx.</math> The sequence of Laguerre polynomials {{math\|''n''! L<sub>''n''</sub>}} is a [[Sheffer sequence]], :<math> \frac{d}{dx} L_n = \left ( \frac{d}{dx} - 1 \right ) L_{n-1}.</math> The [[rook polynomial]]s in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the [[Tricomi–Carlitz polynomials]]. The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the [[Schrödinger equation]] for a one-electron atom. They also describe the static Wigner functions of oscillator systems in [[Phase space formulation#Simple harmonic oscillator\|quantum mechanics in phase space]]. They further enter in the quantum mechanics of the [[Morse potential]] and of the [[Quantum harmonic oscillator#Example: 3D isotropic harmonic oscillator\|3D isotropic harmonic oscillator]]. Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of ''n''<nowiki>!</nowiki> 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: {\| class="wikitable" style="margin:0.5em auto" \|- ! width="20%"\| ''n'' ! <math>L_n(x)\,</math> \|- \| align="center" \| 0 \|\| <math>1\,</math> \|- \| align="center" \| 1 \|\| <math>-x+1\,</math> \|- \| align="center" \| 2 \| <math> \tfrac{1}{2} (x^2-4x+2) \,</math> \|- \| align="center" \| 3 \| <math>\tfrac{1}{6} (-x^3+9x^2-18x+6) \,</math> \|- \| align="center" \| 4 \| <math>\tfrac{1}{24} (x^4-16x^3+72x^2-96x+24) \,</math> \|- \| align="center" \| 5 \| <math>\tfrac{1}{120} (-x^5+25x^4-200x^3+600x^2-600x+120) \,</math> \|- \| align="center" \| 6 \| <math>\tfrac{1}{720} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,</math> \|- \| align="center" \| n \| <math>\tfrac{1}{n!} ((-x)^n + n^2(-x)^{n-1} + ... + n({n!})(-x) + n!) \,</math> \|} [[Image:Laguerre poly.svg\|thumb\|center\|600px\|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 : <math>L_0(x) = 1</math> : <math>L_1(x) = 1 - x</math> and then using the following [[Orthogonal polynomials#Recurrence relations\|recurrence relation]] for any ''k'' ≥ 1: : <math>L_{k + 1}(x) = \frac{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}. </math> In solution of some boundary value problems, the characteristic values can be useful: : <math>L_{k}(0) = 1, L_{k}'(0) = -k. </math> The '''closed form''' is :<math>L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k .</math> The [[generating function]] for them likewise follows, :<math>\sum_{n=0}^\infty t^n L_n(x)= \frac{1}{1-t} e^{-tx/(1-t)}.</math> Polynomials of negative index can be expressed using the ones with positive index: :<math>L_{-n}(x)=e^xL_{n-1}(-x).</math> == Generalized Laguerre polynomials == For arbitrary real α the polynomial solutions of the differential equation<ref>A&S p. 781</ref> :<math>x\,y'' + (\alpha +1 - x)\,y' + n\,y = 0</math> are called '''generalized Laguerre polynomials''', or '''associated Laguerre polynomials'''. One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as : <math>L^{(\alpha)}_0(x) = 1</math> : <math>L^{(\alpha)}_1(x) = 1 + \alpha - x</math> and then using the following [[Orthogonal polynomials#Recurrence relations\|recurrence relation]] for any ''k'' ≥ 1: : <math>L^{(\alpha)}_{k + 1}(x) = \frac{(2k + 1 + \alpha - x)L^{(\alpha)}_k(x) - (k + \alpha) L^{(\alpha)}_{k - 1}(x)}{k + 1}. </math> The simple Laguerre polynomials are the special case {{math\|''α'' {{=}} 0}} of the generalized Laguerre polynomials: : <math>L^{(0)}_n(x)=L_n(x).</math> The [[Rodrigues formula]] for them is :<math>\begin{align} L_n^{(\alpha)}(x) &= {x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right) \\[4pt] &= x^{-\alpha} \frac{\left( \frac{d}{dx}-1\right)^n}{n!}x^{n+\alpha}. \end{align}</math> The [[generating function]] for them is :<math>\sum_{n=0}^\infty t^n L^{(\alpha)}_n(x)= \frac{1}{(1-t)^{\alpha+1}} e^{-tx/(1-t)}.</math> [[File:Zugeordnete Laguerre-Polynome.svg\|thumb\|center\|600px\|The first few generalized Laguerre polynomials, ''L<sub>n</sub>''<sup>(''k'')</sup>(''x'')]] === Explicit examples and properties of the generalized Laguerre polynomials === * Laguerre functions are defined by [[confluent hypergeometric function]]s and Kummer's transformation as<ref>A&S p. 509</ref> ::<math> L_n^{(\alpha)}(x) := {n+ \alpha \choose n} M(-n,\alpha+1,x).</math> :<math>{n+ \alpha \choose n}</math> is a generalized [[binomial coefficient]]. When {{mvar\|n}} is an integer the function reduces to a polynomial of degree {{mvar\|n}}. It has the alternative expression<ref>A&S p. 510</ref> :: <math>L_n^{(\alpha)}(x)= \frac {(-1)^n}{n!} U(-n,\alpha+1,x)</math> :in terms of [[confluent hypergeometric function\|Kummer's function of the second kind]]. * The closed form for these generalized Laguerre polynomials of degree {{mvar\|n}} is<ref>A&S p. 775</ref> :: <math> L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!} </math> :derived by applying [[Leibniz rule (generalized product rule)\|Leibniz's theorem for differentiation of a product]] to Rodrigues' formula. * The first few generalized Laguerre polynomials are: :: <math>\begin{align} L_0^{(\alpha)}(x) &= 1 \\ L_1^{(\alpha)}(x) &= -x + \alpha +1 \\ L_2^{(\alpha)}(x) &= \frac{x^2}{2} - (\alpha + 2)x + \frac{(\alpha+2)(\alpha+1)}{2} \\ L_3^{(\alpha)}(x) &= \frac{-x^3}{6} + \frac{(\alpha+3)x^2}{2} -\frac{(\alpha+2)(\alpha+3)x}{2} +\frac{(\alpha+1)(\alpha+2)(\alpha+3)}{6} \end{align}</math> * The [[coefficient]] of the leading term is (−1)<sup>''n''</sup>/''n''<nowiki>!</nowiki>; * The [[constant term]], which is the value at 0, is :: <math>L_n^{(\alpha)}(0) = {n+\alpha\choose n} = \frac{n^\alpha}{\Gamma(\alpha+1)} + O\left(n^{\alpha-1}\right);</math> * If ''α'' is non-negative, then ''L''<sub>''n''</sub><sup>(''α'')</sup> has ''n'' [[real number\|real]], strictly positive [[Root of a function\|roots]] (notice that <math>\left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n</math> is a [[Sturm chain]]), which are all in the [[Interval (mathematics)\|interval]] <math>\left( 0, n+\alpha+ (n-1) \sqrt{n+\alpha} \, \right].</math>{{citation needed\|date=September 2011}} * The polynomials' asymptotic behaviour for large {{mvar\|n}}, but fixed {{mvar\|α}} and {{math\|''x'' > 0}}, is given by<ref>Szegő, p. 198.</ref><ref>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}}</ref> :: <math> \begin{align} & L_n^{(\alpha)}(x) = \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{\sqrt{\pi}} \frac{e^{\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \sin\left(2 \sqrt{nx}- \frac{\pi}{2}\left(\alpha-\frac{1}{2} \right) \right)+O\left(n^{\frac{\alpha}{2}-\frac{3}{4}}\right), \\[6pt] & L_n^{(\alpha)}(-x) = \frac{(n+1)^{\frac{\alpha}{2}-\frac{1}{4}}}{2\sqrt{\pi}} \frac{e^{-x/2}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} e^{2 \sqrt{x(n+1)}} \cdot\left(1+O\left(\frac{1}{\sqrt{n+1}}\right)\right), \end{align} </math> : and summarizing by :: <math>\frac{L_n^{(\alpha)}\left(\frac x n\right)}{n^\alpha}\approx e^{x/ 2n} \cdot \frac{J_\alpha\left(2\sqrt x\right)}{\sqrt x^\alpha},</math> :where <math>J_\alpha</math> is the [[Bessel function#Asymptotic forms\|Bessel function]]. === As a contour integral === Given the generating function specified above, the polynomials may be expressed in terms of a [[contour integral]] : <math>L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint_C\frac{e^{-xt/(1-t)}}{(1-t)^{\alpha+1}\,t^{n+1}} \; dt,</math> where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1 === Recurrence relations === The addition formula for Laguerre polynomials:<ref>A&S equation (22.12.6), p. 785</ref> : <math>L_n^{(\alpha+\beta+1)}(x+y)= \sum_{i=0}^n L_i^{(\alpha)}(x) L_{n-i}^{(\beta)}(y) </math>. Laguerre's polynomials satisfy the recurrence relations :<math>L_n^{(\alpha)}(x)= \sum_{i=0}^n L_{n-i}^{(\alpha+i)}(y)\frac{(y-x)^i}{i!},</math> in particular : <math>L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x)</math> and : <math>L_n^{(\alpha)}(x)= \sum_{i=0}^n {\alpha-\beta+n-i-1 \choose n-i} L_i^{(\beta)}(x),</math> or : <math>L_n^{(\alpha)}(x)=\sum_{i=0}^n {\alpha-\beta+n \choose n-i} L_i^{(\beta- i)}(x);</math> moreover : <math>\begin{align} L_n^{(\alpha)}(x)- \sum_{j=0}^{\Delta-1} {n+\alpha \choose n-j} (-1)^j \frac{x^j}{j!}&= (-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{{n+\alpha \choose n-\Delta-i}}{(n-i){n \choose i}}L_i^{(\alpha+\Delta)}(x)\\[6pt] &=(-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{{n+\alpha-i-1 \choose n-\Delta-i}}{(n-i){n \choose i}}L_i^{(n+\alpha+\Delta-i)}(x) \end{align}</math> They can be used to derive the four 3-point-rules :<math>\begin{align} L_n^{(\alpha)}(x) &= L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x) = \sum_{j=0}^k {k \choose j} L_{n-j}^{(\alpha+k)}(x), \\[10pt] n L_n^{(\alpha)}(x) &= (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x), \\[10pt] & \text{or } \\ \frac{x^k}{k!}L_n^{(\alpha)}(x) &= \sum_{i=0}^k (-1)^i {n+i \choose i} {n+\alpha \choose k-i} L_{n+i}^{(\alpha-k)}(x), \\[10pt] n L_n^{(\alpha+1)}(x) &= (n-x) L_{n-1}^{(\alpha+1)}(x) + (n+\alpha)L_{n-1}^{(\alpha)}(x) \\[10pt] x L_n^{(\alpha+1)}(x) &= (n+\alpha)L_{n-1}^{(\alpha)}(x)-(n-x)L_n^{(\alpha)}(x); \end{align}</math> combined they give this additional, useful recurrence relations :<math>\begin{align} L_n^{(\alpha)}(x)&= \left(2+\frac{\alpha-1-x}n \right)L_{n-1}^{(\alpha)}(x)- \left(1+\frac{\alpha-1}n \right)L_{n-2}^{(\alpha)}(x)\\[10pt] &= \frac{\alpha+1-x}n L_{n-1}^{(\alpha+1)}(x)- \frac x n L_{n-2}^{(\alpha+2)}(x) \end{align}</math> Since <math>L_n^{(\alpha)}(x)</math> is a monic polynomial of degree <math>n</math> in <math>\alpha</math>, there is the [[partial fraction decomposition]] : <math>\begin{align} \frac{n!\,L_n^{(\alpha)}(x)}{(\alpha+1)_n} &= 1- \sum_{j=1}^n (-1)^j \frac{j}{\alpha + j} {n \choose j}L_n^{(-j)}(x) \\ &= 1- \sum_{j=1}^n \frac{x^j}{\alpha + j}\,\,\frac{L_{n-j}^{(j)}(x)}{(j-1)!} \\ &= 1-x \sum_{i=1}^n \frac{L_{n-i}^{(-\alpha)}(x) L_{i-1}^{(\alpha+1)}(-x)}{\alpha +i}. \end{align}</math> The second equality follows by the following identity, valid for integer ''i'' and {{mvar\|n}} and immediate from the expression of <math>L_n^{(\alpha)}(x)</math> in terms of [[Charlier polynomials]]: : <math> \frac{(-x)^i}{i!} L_n^{(i-n)}(x) = \frac{(-x)^n}{n!} L_i^{(n-i)}(x).</math> 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 : <math>\frac{d^k}{d x^k} L_n^{(\alpha)} (x) = \begin{cases}(-1)^k L_{n-k}^{(\alpha+k)}(x) & \text{if } k\le n, \\ 0 & \text{otherwise.} \end{cases}</math> This points to a special case ({{math\|''α'' {{=}} 0}}) of the formula above: for integer {{math\|''α'' {{=}} ''k''}} the generalized polynomial may be written :<math>L_n^{(k)}(x)=(-1)^k\frac{d^kL_{n+k}(x)}{dx^k},</math> the shift by ''k'' sometimes causing confusion with the usual parenthesis notation for a derivative. Moreover, the following equation holds: : <math>\frac{1}{k!} \frac{d^k}{d x^k} x^\alpha L_n^{(\alpha)} (x) = {n+\alpha \choose k} x^{\alpha-k} L_n^{(\alpha-k)}(x),</math> which generalizes with [[Antiderivative#Techniques of integration\|Cauchy's formula]] to : <math>L_n^{(\alpha')}(x) = (\alpha'-\alpha) {\alpha'+ n \choose \alpha'-\alpha} \int_0^x \frac{t^\alpha (x-t)^{\alpha'-\alpha-1}}{x^{\alpha'}} L_n^{(\alpha)}(t)\,dt.</math> The derivative with respect to the second variable {{mvar\|α}} has the form,<ref>{{Cite journal \| doi=10.1080/10652469708819127\|title = Identities for families of orthogonal polynomials and special functions\| journal=Integral Transforms and Special Functions\| volume=5\| issue=1–2\| pages=69–102\|year = 1997\|last1 = Koepf\|first1 = Wolfram\| citeseerx=10.1.1.298.7657}}</ref> : <math>\frac{d}{d \alpha}L_n^{(\alpha)}(x)= \sum_{i=0}^{n-1} \frac{L_i^{(\alpha)}(x)}{n-i}.</math> This is evident from the contour integral representation below. The generalized Laguerre polynomials obey the differential equation : <math>x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0,</math> which may be compared with the equation obeyed by the ''k''th derivative of the ordinary Laguerre polynomial, : <math>x L_n^{[k] \prime\prime}(x) + (k+1-x)L_n^{[k]\prime}(x) + (n-k) L_n^{[k]}(x)=0,</math> where <math>L_n^{[k]}(x)\equiv\frac{d^kL_n(x)}{dx^k}</math> for this equation only. In [[Sturm–Liouville theory\|Sturm–Liouville form]] the differential equation is : <math>-\left(x^{\alpha+1} e^{-x}\cdot L_n^{(\alpha)}(x)^\prime\right)^\prime= n\cdot x^\alpha e^{-x}\cdot L_n^{(\alpha)}(x),</math> which shows that {{math\|''L''{{su\|b=''n''\|p=''(α)''}}}} is an eigenvector for the eigenvalue {{mvar\|n}}. === Orthogonality === The generalized Laguerre polynomials are orthogonal over {{math\|[0, ∞)}} with respect to the measure with weighting function {{math\|''x<sup>α</sup>'' ''e''<sup>−''x''</sup>}}:<ref>{{Cite web \| url=http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html \| title=Associated Laguerre Polynomial}}</ref> :<math>\int_0^\infty x^\alpha e^{-x} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!} \delta_{n,m},</math> which follows from :<math>\int_0^\infty x^{\alpha'-1} e^{-x} L_n^{(\alpha)}(x)dx= {\alpha-\alpha'+n \choose n} \Gamma(\alpha').</math> If <math>\Gamma(x,\alpha+1,1)</math> denotes the Gamma distribution then the orthogonality relation can be written as :<math>\int_0^{\infty} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)\Gamma(x,\alpha+1,1) dx={n+ \alpha \choose n}\delta_{n,m},</math> The associated, symmetric kernel polynomial has the representations ([[Christoffel–Darboux formula]]){{citation needed\|date=October 2011}}<!--All of these formulas require citations.--> :<math>\begin{align} K_n^{(\alpha)}(x,y) &:= \frac{1}{\Gamma(\alpha+1)} \sum_{i=0}^n \frac{L_i^{(\alpha)}(x) L_i^{(\alpha)}(y)}{{\alpha+i \choose i}}\\[4pt] & =\frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha)}(x) L_{n+1}^{(\alpha)}(y) - L_{n+1}^{(\alpha)}(x) L_n^{(\alpha)}(y)}{\frac{x-y}{n+1} {n+\alpha \choose n}} \\[4pt] &= \frac{1}{\Gamma(\alpha+1)}\sum_{i=0}^n \frac{x^i}{i!} \frac{L_{n-i}^{(\alpha+i)}(x) L_{n-i}^{(\alpha+i+1)}(y)}{{\alpha+n \choose n}{n \choose i}}; \end{align}</math> recursively : <math>K_n^{(\alpha)}(x,y)=\frac{y}{\alpha+1} K_{n-1}^{(\alpha+1)}(x,y)+ \frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha+1)}(x) L_n^{(\alpha)}(y)}{{\alpha+n \choose n}}.</math> Moreover,{{clarify\|post-text=Limit as n goes to infinity?\|date=January 2016}} : <math>y^\alpha e^{-y} K_n^{(\alpha)}(\cdot, y) \to \delta(y- \cdot).</math> [[Turán's inequalities]] can be derived here, which is : <math>L_n^{(\alpha)}(x)^2- L_{n-1}^{(\alpha)}(x) L_{n+1}^{(\alpha)}(x)= \sum_{k=0}^{n-1} \frac{{\alpha+n-1\choose n-k}}{n{n\choose k}} L_k^{(\alpha-1)}(x)^2>0.</math> The following integral is needed in the quantum mechanical treatment of the [[Hydrogen atom#Wavefunction\|hydrogen atom]], : <math>\int_0^{\infty}x^{\alpha+1} e^{-x} \left[L_n^{(\alpha)} (x)\right]^2 dx= \frac{(n+\alpha)!}{n!}(2n+\alpha+1).</math> === Series expansions === Let a function have the (formal) series expansion : <math>f(x)= \sum_{i=0}^\infty f_i^{(\alpha)} L_i^{(\alpha)}(x).</math> Then : <math>f_i^{(\alpha)}=\int_0^\infty \frac{L_i^{(\alpha)}(x)}{{i+ \alpha \choose i}} \cdot \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} \cdot f(x) \,dx .</math> The series converges in the associated [[Hilbert space]] [[Lp space\|{{math\|''L''<sup>2</sup>[0, ∞)}}]] [[if and only if]] : <math>\\| f \\|_{L^2}^2 := \int_0^\infty \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} \| f(x)\|^2 \, dx = \sum_{i=0}^\infty {i+\alpha \choose i} \|f_i^{(\alpha)}\|^2 < \infty. </math> ==== Further examples of expansions==== [[Monomial]]s are represented as : <math>\frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x),</math> while [[binomial coefficient\|binomials]] have the parametrization : <math>{n+x \choose n}= \sum_{i=0}^n \frac{\alpha^i}{i!} L_{n-i}^{(x+i)}(\alpha).</math> This leads directly to : <math>e^{-\gamma x}= \sum_{i=0}^\infty \frac{\gamma^i}{(1+\gamma)^{i+\alpha+1}} L_i^{(\alpha)}(x) \qquad \text{convergent iff } \Re(\gamma) > -\tfrac{1}{2}</math> for the exponential function. The [[incomplete gamma function]] has the representation : <math>\Gamma(\alpha,x)=x^\alpha e^{-x} \sum_{i=0}^\infty \frac{L_i^{(\alpha)}(x)}{1+i} \qquad \left(\Re(\alpha)>-1 , x > 0\right).</math> ==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.<ref>{{Cite book\|title=Quantum Mechanics in Chemistry\|last=Ratner, Schatz\|first=Mark A., George C.\|publisher=Prentice Hall\|year=2001\|isbn=\|location=0-13-895491-7\|pages=90–91}}</ref> [[Vibronic coupling\|Vibronic transitions]] in the Franck-Condon approximation can also be described using Laguerre polynomials.<ref>{{Cite journal\|last=Jong\|first=Mathijs de\|last2=Seijo\|first2=Luis\|last3=Meijerink\|first3=Andries\|last4=Rabouw\|first4=Freddy T.\|date=2015-06-24\|title=Resolving the ambiguity in the relation between Stokes shift and Huang–Rhys parameter\|url=https://pubs.rsc.org/en/content/articlelanding/2015/cp/c5cp02093j\|journal=Physical Chemistry Chemical Physics\|language=en\|volume=17\|issue=26\|pages=16959–16969\|doi=10.1039/C5CP02093J\|issn=1463-9084}}</ref> ==Multiplication theorems== [[Arthur Erdélyi\|Erdélyi]] gives the following two [[multiplication theorem]]s <ref>C. Truesdell, "[http://www.pnas.org/cgi/reprint/36/12/752.pdf On the Addition and Multiplication Theorems for the Special Functions]", ''Proceedings of the National Academy of Sciences, Mathematics'', (1950) pp. 752–757.</ref> :<math> \begin{align} & t^{n+1+\alpha} e^{(1-t) z} L_n^{(\alpha)}(z t)=\sum_{k=n}^\infty {k \choose n}\left(1-\frac 1 t\right)^{k-n} L_k^{(\alpha)}(z), \\[6pt] & e^{(1-t)z} L_n^{(\alpha)}(z t)=\sum_{k=0}^\infty \frac{(1-t)^k z^k}{k!}L_n^{(\alpha+k)}(z). \end{align} </math> == Relation to Hermite polynomials == The generalized Laguerre polynomials are related to the [[Hermite polynomial]]s: : <math>\begin{align} H_{2n}(x) &= (-1)^n 2^{2n} n! L_n^{(-1/2)} (x^2) \\[4pt] H_{2n+1}(x) &= (-1)^n 2^{2n+1} n! x L_n^{(1/2)} (x^2) \end{align}</math> where the ''H''<sub>''n''</sub>(''x'') are the [[Hermite polynomial]]s based on the weighting function exp(−''x''<sup>2</sup>), the so-called "physicist's version." Because of this, the generalized Laguerre polynomials arise in the treatment of the [[quantum harmonic oscillator]]. == Relation to hypergeometric functions == The Laguerre polynomials may be defined in terms of [[hypergeometric function]]s, specifically the [[confluent hypergeometric function]]s, as : <math>L^{(\alpha)}_n(x) = {n+\alpha \choose n} M(-n,\alpha+1,x) =\frac{(\alpha+1)_n} {n!} \,_1F_1(-n,\alpha+1,x)</math> where <math>(a)_n</math> is the [[Pochhammer symbol]] (which in this case represents the rising factorial). == Hardy–Hille formula == The generalized Laguerre polynomials satisfy the Hardy–Hille formula<ref>Szegő, p. 102.</ref><ref>W. A. Al-Salam (1964), [https://projecteuclid.org/euclid.dmj/1077375084 "Operational representations for Laguerre and other polynomials"], ''Duke Math J.'' '''31''' (1): 127–142.</ref> : <math>\sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right),</math> where the series on the left converges for <math>\alpha>-1</math> and <math>\|t\|<1</math>. Using the identity : <math>\,_0F_1(;\alpha + 1;z)=\,\Gamma(\alpha + 1) z^{-\alpha/2} I_\alpha\left(2\sqrt{z}\right),</math> (see [[Generalized hypergeometric function#The series 0F1\|generalized hypergeometric function]]), this can also be written as : <math>\sum_{n=0}^\infty \frac{n!}{\Gamma(1+\alpha+n)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y) t^n = \frac{1}{(xyt)^{\alpha/2}(1-t)}e^{-(x+y)t/(1-t)} I_\alpha \left(\frac{2\sqrt{xyt}}{1-t}\right).</math> This formula is a generalization of the [[Mehler kernel]] for [[Hermite polynomial]]s, which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above. == See also == * [[Angelescu polynomials]] * [[Transverse mode]], an important application of Laguerre polynomials to describe the field intensity within a waveguide or laser beam profile. == Notes == {{Reflist\|35em}} == References == * {{Abramowitz_Stegun_ref\|22\|773}} * G. Szegő, ''Orthogonal polynomials'', 4th edition, ''Amer. Math. Soc. Colloq. Publ.'', vol. 23, Amer. Math. Soc., Providence, RI, 1975. * {{dlmf\|id=18\|title=Orthogonal Polynomials\|first=Tom H.\|last=Koornwinder\|first2=Roderick S. C.\|last2= Wong\|first3=Roelof \| last3=Koekoek \|\|first4=René F. \|last4=Swarttouw}} * B. Spain, M.G. Smith, ''Functions of mathematical physics'', Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials. * {{springer\|title=Laguerre polynomials\|id=p/l057310}} * [[Eric W. Weisstein]], "[http://mathworld.wolfram.com/LaguerrePolynomial.html Laguerre Polynomial]", From MathWorld—A Wolfram Web Resource. * {{cite book\|author=[[George Arfken]] and Hans Weber\| title= Mathematical Methods for Physicists\| publisher=Academic Press\| year=2000\| isbn = 978-0-12-059825-0 }} == External links == * {{cite web\|author=Timothy Jones\|url=http://www.physics.drexel.edu/~tim/open/hydrofin \| title=The Legendre and Laguerre Polynomials and the elementary quantum mechanical model of the Hydrogen Atom}} * {{MathWorld\|title=Laguerre polynomial\|id=LaguerrePolynomial}} [[Category:Polynomials]] [[Category:Orthogonal polynomials]] [[Category:Special hypergeometric functions]]
	Purge cached results.

LaTeX to CAS translator

Mathematica

Information

Tests

Symbolic

Numeric

SymPy

Information

Tests

Symbolic

Numeric

Maple

Information

Tests

Symbolic

Numeric

Dependency Graph Information

Specify your own input

Navigation menu

Search