Laguerre polynomials

One can also define the Laguerre polynomials recursively, defining the first two polynomials as

$$L\_0(x)\; =\; 1$$

$$L\_1(x)\; =\; 1\; -\; x$$

and then using the following recurrence relation for any {{math|k ≥ 1}}:

$$L\_\{k\; +\; 1\}(x)\; =\; \backslash frac\{(2k\; +\; 1\; -\; x)L\_k(x)\; -\; k\; L\_\{k\; -\; 1\}(x)\}\{k\; +\; 1\}.$$

Furthermore,

$$x\; L\text{'}\_n(x)\; =\; nL\_n\; (x)\; -\; nL\_\{n-1\}(x).$$

In solution of some boundary value problems, the characteristic values can be useful:

$$L\_\{k\}(0)\; =\; 1,\; L\_\{k\}\text{'}(0)\; =\; -k.$$

The closed form is

$$L\_n(x)=\backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\backslash frac\{(-1)^k\}\{k!\}\; x^k\; .$$

The generating function for them likewise follows,

$$\backslash sum\_\{n=0\}^\backslash infty\; t^n\; L\_n(x)=\; \backslash frac\{1\}\{1-t\}\; e^\{-tx/(1-t)\}.$$The operator form is

$$L\_n(x)\; =\; \backslash frac\{1\}\{n!\}e^x\; \backslash frac\{d^n\}\{dx^n\}\; (x^n\; e^\{-x\})$$

Polynomials of negative index can be expressed using the ones with positive index:

$$L\_\{-n\}(x)=e^xL\_\{n-1\}(-x).$$

Image:Laguerre poly.svg

For arbitrary real α the polynomial solutions of the differential equationA&S p. 781

$$x\backslash ,y\text{'}\text{'}\; +\; \backslash left(\backslash alpha\; +1\; -\; x\backslash right)\; y\text{'}\; +\; n\backslash ,y\; =\; 0$$

are called generalized Laguerre polynomials, or associated Laguerre polynomials.

One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as

$$L^\{(\backslash alpha)\}\_0(x)\; =\; 1$$

$$L^\{(\backslash alpha)\}\_1(x)\; =\; 1\; +\; \backslash alpha\; -\; x$$

and then using the following recurrence relation for any {{math|k ≥ 1}}:

$$L^\{(\backslash alpha)\}\_\{k\; +\; 1\}(x)\; =\; \backslash frac\{(2k\; +\; 1\; +\; \backslash alpha\; -\; x)L^\{(\backslash alpha)\}\_k(x)\; -\; (k\; +\; \backslash alpha)\; L^\{(\backslash alpha)\}\_\{k\; -\; 1\}(x)\}\{k\; +\; 1\}.$$

The simple Laguerre polynomials are the special case {{math|1=α = 0}} of the generalized Laguerre polynomials:

$$L^\{(0)\}\_n(x)\; =\; L\_n(x).$$

The Rodrigues formula for them is

$$L\_n^\{(\backslash alpha)\}(x)\; =\; \{x^\{-\backslash alpha\}\; e^x\; \backslash over\; n!\}\{d^n\; \backslash over\; dx^n\}\; \backslash left(e^\{-x\}\; x^\{n+\backslash alpha\}\backslash right)$$

= \frac{x^{-\alpha}}{n!}\left( \frac{d}{dx}-1\right)^nx^{n+\alpha}.

The generating function for them is

$$\backslash sum\_\{n=0\}^\backslash infty\; t^n\; L^\{(\backslash alpha)\}\_n(x)=\; \backslash frac\{1\}\{(1-t)^\{\backslash alpha+1\}\}\; e^\{-tx/(1-t)\}.$$

File:Zugeordnete Laguerre-Polynome.svg

• Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation asA&S p. 509 $$L\_n^\{(\backslash alpha)\}(x)\; :=\; \{n+\; \backslash alpha\; \backslash choose\; n\}\; M(-n,\backslash alpha+1,x).$$ where $\{n+\; \backslash alpha\; \backslash choose\; n\}$ 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 expressionA&S p. 510 $$L\_n^\{(\backslash alpha)\}(x)=\; \backslash frac\; \{(-1)^n\}\{n!\}\; U(-n,\backslash alpha+1,x)$$ in terms of Kummer's function of the second kind.
• The closed form for these generalized Laguerre polynomials of degree {{mvar|n}} isA&S p. 775 $$L\_n^\{(\backslash alpha)\}\; (x)\; =\; \backslash sum\_\{i=0\}^n\; (-1)^i\; \{n+\backslash alpha\; \backslash choose\; n-i\}\; \backslash frac\{x^i\}\{i!\}$$ derived by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.
• Laguerre polynomials have a differential operator representation, much like the closely related Hermite polynomials. Namely, let $D\; =\; \backslash frac\{d\}\{dx\}$ and consider the differential operator $M=xD^2+(\backslash alpha+1)D$. Then $\backslash exp(-tM)x^n=(-1)^nt^nn!L^\{(\backslash alpha)\}\_n\backslash left(\backslash frac\{x\}\{t\}\backslash right)$.{{citation needed|date=April 2023}}
• The first few generalized Laguerre polynomials are:

• The coefficient of the leading term is {{math|(−1)ⁿ/n!}};
• The constant term, which is the value at 0, is $$L\_n^\{(\backslash alpha)\}(0)\; =\; \{n+\backslash alpha\backslash choose\; n\}\; =\; \backslash frac\{\backslash Gamma(n\; +\; \backslash alpha\; +\; 1)\}\{n!\backslash ,\; \backslash Gamma(\backslash alpha\; +\; 1)\};$$

• If {{math|α}} is non-negative, then L_n^(α) has n real, strictly positive roots (notice that $\backslash left((-1)^\{n-i\}\; L\_\{n-i\}^\{(\backslash alpha)\}\backslash right)\_\{i=0\}^n$ is a Sturm chain), which are all in the interval $\backslash left(\; 0,\; n+\backslash alpha+\; (n-1)\; \backslash sqrt\{n+\backslash alpha\}\; \backslash ,\; \backslash right].${{citation needed|date=September 2011}}
• The polynomials' asymptotic behaviour for large {{mvar|n}}, but fixed {{mvar|α}} and {{math|x > 0}}, is given bySzegő, p. 198.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}} $$$$

\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}

and summarizing by $$\backslash frac\{L\_n^\{(\backslash alpha)\}\backslash left(\backslash frac\; x\; n\backslash right)\}\{n^\backslash alpha\}\backslash approx\; e^\{x/\; 2n\}\; \backslash cdot\; \backslash frac\{J\_\backslash alpha\backslash left(2\backslash sqrt\; x\backslash right)\}\{\backslash sqrt\; x^\backslash alpha\},$$ where $J\_\backslash alpha$ is the Bessel function.

Given the generating function specified above, the polynomials may be expressed in terms of a contour integral

$$L\_n^\{(\backslash alpha)\}(x)=\backslash frac\{1\}\{2\backslash pi\; i\}\backslash oint\_C\backslash frac\{e^\{-xt/(1-t)\}\}\{(1-t)^\{\backslash alpha+1\}\backslash ,t^\{n+1\}\}\; \backslash ;\; dt,$$

where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1

The addition formula for Laguerre polynomials:{{Cite web |title=DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials |url=https://dlmf.nist.gov/18.18 |access-date=2025-03-18 |website=dlmf.nist.gov}}

$$L^\{(\backslash alpha\_\{1\}+\backslash dots+\backslash alpha\_\{r\}+r-1)\}\_\{n\}\backslash left(x\_\{1\}+\backslash dots+x\_\{r\}\backslash right)=\backslash sum\_\{m\_\{1\}+\backslash dots+m\_\{r\}=n\}L^\{(\backslash alpha\_\{1\})\}\_\{m\_\{1\}\}\backslash left(x\_\{1\}\backslash right)\backslash cdots\; L^\{(\backslash alpha\_\{r\})\}\_\{m\_\{r\}\}\backslash left(x\_\{r\}\backslash right).$$Laguerre's polynomials satisfy the recurrence relations

$$L\_n^\{(\backslash alpha)\}(x)=\; \backslash sum\_\{i=0\}^n\; L\_\{n-i\}^\{(\backslash alpha+i)\}(y)\backslash frac\{(y-x)^i\}\{i!\},$$

in particular

$$L\_n^\{(\backslash alpha+1)\}(x)=\; \backslash sum\_\{i=0\}^n\; L\_i^\{(\backslash alpha)\}(x)$$

and

$$L\_n^\{(\backslash alpha)\}(x)=\; \backslash sum\_\{i=0\}^n\; \{\backslash alpha-\backslash beta+n-i-1\; \backslash choose\; n-i\}\; L\_i^\{(\backslash beta)\}(x),$$

or

$$L\_n^\{(\backslash alpha)\}(x)=\backslash sum\_\{i=0\}^n\; \{\backslash alpha-\backslash beta+n\; \backslash choose\; n-i\}\; L\_i^\{(\backslash beta-\; i)\}(x);$$

moreover

$$\backslash begin\{align\}$$

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}

They can be used to derive the four 3-point-rules

$$\backslash begin\{align\}$$

L_n^{(\alpha)}(x) &= L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x) = \sum_{j=0}^k {k \choose j}(-1)^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}

combined they give this additional, useful recurrence relations$$\backslash begin\{align\}$$

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}

Since $L\_n^\{(\backslash alpha)\}(x)$ is a monic polynomial of degree $n$ in $\backslash alpha$,

there is the partial fraction decomposition

$$\backslash begin\{align\}$$

\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}

The second equality follows by the following identity, valid for integer i and {{mvar|n}} and immediate from the expression of $L\_n^\{(\backslash alpha)\}(x)$ in terms of Charlier polynomials:

$$\backslash frac\{(-x)^i\}\{i!\}\; L\_n^\{(i-n)\}(x)\; =\; \backslash frac\{(-x)^n\}\{n!\}\; L\_i^\{(n-i)\}(x).$$

For the third equality apply the fourth and fifth identities of this section.

Differentiating the power series representation of a generalized Laguerre polynomial {{mvar|k}} times leads to

$$\backslash frac\{d^k\}\{d\; x^k\}\; L\_n^\{(\backslash alpha)\}\; (x)\; =\; \backslash begin\{cases\}$$

(-1)^k L_{n-k}^{(\alpha+k)}(x) & \text{if } k\le n, \\

0 & \text{otherwise.}

\end{cases}

This points to a special case ({{math|1=α = 0}}) of the formula above: for integer {{math|1=α = k}} the generalized polynomial may be written

$$L\_n^\{(k)\}(x)=(-1)^k\backslash frac\{d^kL\_\{n+k\}(x)\}\{dx^k\},$$

the shift by {{mvar|k}} sometimes causing confusion with the usual parenthesis notation for a derivative.

Moreover, the following equation holds:

$$\backslash frac\{1\}\{k!\}\; \backslash frac\{d^k\}\{d\; x^k\}\; x^\backslash alpha\; L\_n^\{(\backslash alpha)\}\; (x)\; =\; \{n+\backslash alpha\; \backslash choose\; k\}\; x^\{\backslash alpha-k\}\; L\_n^\{(\backslash alpha-k)\}(x),$$

which generalizes with Cauchy's formula to

$$L\_n^\{(\backslash alpha\text{'})\}(x)\; =\; (\backslash alpha\text{'}-\backslash alpha)\; \{\backslash alpha\text{'}+\; n\; \backslash choose\; \backslash alpha\text{'}-\backslash alpha\}\; \backslash int\_0^x\; \backslash frac\{t^\backslash alpha\; (x-t)^\{\backslash alpha\text{'}-\backslash alpha-1\}\}\{x^\{\backslash alpha\text{'}\}\}\; L\_n^\{(\backslash alpha)\}(t)\backslash ,dt.$$

The derivative with respect to the second variable {{mvar|α}} has the form,{{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}}

$$\backslash frac\{d\}\{d\; \backslash alpha\}L\_n^\{(\backslash alpha)\}(x)=\; \backslash sum\_\{i=0\}^\{n-1\}\; \backslash frac\{L\_i^\{(\backslash alpha)\}(x)\}\{n-i\}.$$

The generalized Laguerre polynomials obey the differential equation

$$x\; L\_n^\{(\backslash alpha)\; \backslash prime\backslash prime\}(x)\; +\; (\backslash alpha+1-x)L\_n^\{(\backslash alpha)\backslash prime\}(x)\; +\; n\; L\_n^\{(\backslash alpha)\}(x)=0,$$

which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,

$$x\; L\_n^\{[k]\; \backslash prime\backslash prime\}(x)\; +\; (k+1-x)L\_n^\{[k]\backslash prime\}(x)\; +\; (n-k)\; L\_n^\{[k]\}(x)=0,$$

where $L\_n^\{[k]\}(x)\backslash equiv\backslash frac\{d^kL\_n(x)\}\{dx^k\}$ for this equation only.

In Sturm–Liouville form the differential equation is

$$-\backslash left(x^\{\backslash alpha+1\}\; e^\{-x\}\backslash cdot\; L\_n^\{(\backslash alpha)\}(x)^\backslash prime\backslash right)\text{'}\; =\; n\backslash cdot\; x^\backslash alpha\; e^\{-x\}\backslash cdot\; L\_n^\{(\backslash alpha)\}(x),$$

which shows that {{math|L{{su|b=n|p=(α)}}}} is an eigenvector for the eigenvalue {{mvar|n}}.

The generalized Laguerre polynomials are orthogonal over {{closed-open|0, ∞}} with respect to the measure with weighting function {{math|x^α e^−x}}:{{Cite web | url=http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html | title=Associated Laguerre Polynomial}}

$$\backslash int\_0^\backslash infty\; x^\backslash alpha\; e^\{-x\}\; L\_n^\{(\backslash alpha)\}(x)L\_m^\{(\backslash alpha)\}(x)dx=\backslash frac\{\backslash Gamma(n+\backslash alpha+1)\}\{n!\}\; \backslash delta\_\{n,m\},$$

which follows from

$$\backslash int\_0^\backslash infty\; x^\{\backslash alpha\text{'}-1\}\; e^\{-x\}\; L\_n^\{(\backslash alpha)\}(x)dx=\; \{\backslash alpha-\backslash alpha\text{'}+n\; \backslash choose\; n\}\; \backslash Gamma(\backslash alpha\text{'}).$$

If $\backslash Gamma(x,\backslash alpha+1,1)$ denotes the gamma distribution then the orthogonality relation can be written as

$$\backslash int\_0^\{\backslash infty\}\; L\_n^\{(\backslash alpha)\}(x)L\_m^\{(\backslash alpha)\}(x)\backslash Gamma(x,\backslash alpha+1,1)\; dx=\{n+\; \backslash alpha\; \backslash choose\; n\}\backslash delta\_\{n,m\}.$$

The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula){{citation needed|date=October 2011}}

$$\backslash begin\{align\}$$

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}

recursively

$$K\_n^\{(\backslash alpha)\}(x,y)=\backslash frac\{y\}\{\backslash alpha+1\}\; K\_\{n-1\}^\{(\backslash alpha+1)\}(x,y)+\; \backslash frac\{1\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash frac\{L\_n^\{(\backslash alpha+1)\}(x)\; L\_n^\{(\backslash alpha)\}(y)\}\{\{\backslash alpha+n\; \backslash choose\; n\}\}.$$

Moreover,{{clarify|post-text=Limit as n goes to infinity?|date=January 2016}}

$$y^\backslash alpha\; e^\{-y\}\; K\_n^\{(\backslash alpha)\}(\backslash cdot,\; y)\; \backslash to\; \backslash delta(y-\; \backslash cdot).$$

Turán's inequalities can be derived here, which is

$$L\_n^\{(\backslash alpha)\}(x)^2-\; L\_\{n-1\}^\{(\backslash alpha)\}(x)\; L\_\{n+1\}^\{(\backslash alpha)\}(x)=\; \backslash sum\_\{k=0\}^\{n-1\}\; \backslash frac\{\{\backslash alpha+n-1\backslash choose\; n-k\}\}\{n\{n\backslash choose\; k\}\}\; L\_k^\{(\backslash alpha-1)\}(x)^2>0.$$

The following integral is needed in the quantum mechanical treatment of the hydrogen atom,

$$\backslash int\_0^\{\backslash infty\}x^\{\backslash alpha+1\}\; e^\{-x\}\; \backslash left[L\_n^\{(\backslash alpha)\}\; (x)\backslash right]^2\; dx=\; \backslash frac\{(n+\backslash alpha)!\}\{n!\}(2n+\backslash alpha+1).$$

Let a function have the (formal) series expansion

$$f(x)=\; \backslash sum\_\{i=0\}^\backslash infty\; f\_i^\{(\backslash alpha)\}\; L\_i^\{(\backslash alpha)\}(x).$$

Then

$$f\_i^\{(\backslash alpha)\}=\backslash int\_0^\backslash infty\; \backslash frac\{L\_i^\{(\backslash alpha)\}(x)\}\{\{i+\; \backslash alpha\; \backslash choose\; i\}\}\; \backslash cdot\; \backslash frac\{x^\backslash alpha\; e^\{-x\}\}\{\backslash Gamma(\backslash alpha+1)\}\; \backslash cdot\; f(x)\; \backslash ,dx\; .$$

The series converges in the associated Hilbert space {{math|L²[0, ∞)}} if and only if

Monomials are represented as

$$\backslash frac\{x^n\}\{n!\}=\; \backslash sum\_\{i=0\}^n\; (-1)^i\; \{n+\; \backslash alpha\; \backslash choose\; n-i\}\; L\_i^\{(\backslash alpha)\}(x),$$

while binomials have the parametrization

$$\{n+x\; \backslash choose\; n\}=\; \backslash sum\_\{i=0\}^n\; \backslash frac\{\backslash alpha^i\}\{i!\}\; L\_\{n-i\}^\{(x+i)\}(\backslash alpha).$$

This leads directly to

$$e^\{-\backslash gamma\; x\}=\; \backslash sum\_\{i=0\}^\backslash infty\; \backslash frac\{\backslash gamma^i\}\{(1+\backslash gamma)^\{i+\backslash alpha+1\}\}\; L\_i^\{(\backslash alpha)\}(x)\; \backslash qquad\; \backslash text\{convergent\; iff\; \}\; \backslash Re(\backslash gamma)\; >\; -\backslash tfrac\{1\}\{2\}$$

for the exponential function. The incomplete gamma function has the representation

$$\backslash Gamma(\backslash alpha,x)=x^\backslash alpha\; e^\{-x\}\; \backslash sum\_\{i=0\}^\backslash infty\; \backslash frac\{L\_i^\{(\backslash alpha)\}(x)\}\{1+i\}\; \backslash qquad\; \backslash left(\backslash Re(\backslash alpha)>-1\; ,\; x\; >\; 0\backslash right).$$

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.{{Cite book|title=Quantum Mechanics in Chemistry|last=Ratner, Schatz|first=Mark A., George C.|publisher=Prentice Hall|year=2001|location=0-13-895491-7| pages=90–91}}

Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.{{Cite journal|last1=Jong|first1=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|pmid=26062123|bibcode=2015PCCP...1716959D|hdl=1874/321453|s2cid=34490576 | issn=1463-9084|hdl-access=free}}

Erdélyi gives the following two multiplication theorems 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.

$$\backslash begin\{align\}$$

& 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}

The generalized Laguerre polynomials are related to the Hermite polynomials:

$$\backslash begin\{align\}$$

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}

where the {{math|H_n(x)}} are the Hermite polynomials based on the weighting function {{math|exp(−x²)}}, the so-called "physicist's version."

Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.

Applying the addition formula,$$(-1)^n\; 2^\{2n\}\; n!\; \backslash ,\; L^\{\backslash left(\backslash frac\{r\}\{2\}-1\backslash right)\}\_\{n\}\backslash Bigl(z\_1^2+\backslash cdots+z\_r^2\backslash Bigr)$$

=\sum_{m_1+\cdots+m_r=n} \prod_{i=1}^r H_{2m_i}(z_i).

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as

$$L^\{(\backslash alpha)\}\_n(x)\; =\; \{n+\backslash alpha\; \backslash choose\; n\}\; M(-n,\backslash alpha+1,x)\; =\backslash frac\{(\backslash alpha+1)\_n\}\; \{n!\}\; \backslash ,\_1F\_1(-n,\backslash alpha+1,x)$$

where $(a)\_n$ is the Pochhammer symbol (which in this case represents the rising factorial).

The generalized Laguerre polynomials satisfy the Hardy–Hille formulaSzegő, p. 102.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.

$$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{n!\backslash ,\backslash Gamma\backslash left(\backslash alpha\; +\; 1\backslash right)\}\{\backslash Gamma\backslash left(n+\backslash alpha+1\backslash right)\}L\_n^\{(\backslash alpha)\}(x)L\_n^\{(\backslash alpha)\}(y)t^n=\backslash frac\{1\}\{(1-t)^\{\backslash alpha\; +\; 1\}\}e^\{-(x+y)t/(1-t)\}\backslash ,\_0F\_1\backslash left(;\backslash alpha\; +\; 1;\backslash frac\{xyt\}\{(1-t)^2\}\backslash right),$$

where the series on the left converges for $\backslash alpha>-1$ and . Using the identity

$$\backslash ,\_0F\_1(;\backslash alpha\; +\; 1;z)=\backslash ,\backslash Gamma(\backslash alpha\; +\; 1)\; z^\{-\backslash alpha/2\}\; I\_\backslash alpha\backslash left(2\backslash sqrt\{z\}\backslash right),$$

(see generalized hypergeometric function), this can also be written as

$$\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{n!\}\{\backslash Gamma(1+\backslash alpha+n)\}L\_n^\{(\backslash alpha)\}(x)L\_n^\{(\backslash alpha)\}(y)\; t^n\; =\; \backslash frac\{1\}\{(xyt)^\{\backslash alpha/2\}(1-t)\}e^\{-(x+y)t/(1-t)\}\; I\_\backslash alpha\; \backslash left(\backslash frac\{2\backslash sqrt\{xyt\}\}\{1-t\}\backslash right).$$where $I\_\backslash alpha$ denotes the modified Bessel function of the first kind, defined as$$$$

I_\alpha(z) = \sum_{k=0}^\infty \frac{1}{k!\, \Gamma(k+\alpha+1)} \left(\frac{z}{2}\right)^{2k+\alpha}

This formula is a generalization of the Mehler kernel for Hermite polynomials, which can be recovered from it by setting the Hermite polynomials as a special case of the associated Laguerre polynomials.

Substitute $t\; \backslash mapsto\; -t/y$ and take the $y\; \backslash to\; \backslash infty$ limit, we obtain Szegő, page 102, Equation (5.1.16) $$$$

\sum_{n=0}^\infty

\frac{t^n}{\Gamma(n+1+\alpha)} L_n^{(\alpha)}(x)

=

\frac{e^t}{(-xt)^{\alpha/2}}I_{\alpha}(2\sqrt{-xt}).

The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals.{{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}}{{cite book |last1=Sakurai |first1=J. J. |title=Modern quantum mechanics |date=2011 |publisher=Addison-Wesley |location=Boston |isbn=978-0805382914 |edition=2nd}}{{cite book |last1=Merzbacher |first1=Eugen |title=Quantum mechanics |date=1998 |publisher=Wiley |location=New York |isbn=0471887021 |edition=3rd}} The convention used throughout this article expresses the generalized Laguerre polynomials as {{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}}

$$L\_n^\{(\backslash alpha)\}(x)\; =\; \backslash frac\{\backslash Gamma(\backslash alpha\; +\; n\; +\; 1)\}\{\backslash Gamma(\backslash alpha\; +\; 1)\; n!\}\; \backslash ,\_1F\_1(-n;\; \backslash alpha\; +\; 1;\; x),$$

where $\backslash ,\_1F\_1(a;b;x)$ is the confluent hypergeometric function.

In the physics literature, the generalized Laguerre polynomials are instead defined as

$$\backslash bar\{L\}\_n^\{(\backslash alpha)\}(x)\; =\; \backslash frac\{\backslash left[\backslash Gamma(\backslash alpha\; +\; n\; +\; 1)\backslash right]^2\}\{\backslash Gamma(\backslash alpha\; +\; 1)n!\}\; \backslash ,\_1F\_1(-n;\; \backslash alpha\; +\; 1;\; x).$$

The physics version is related to the standard version by

$$\backslash bar\{L\}\_n^\{(\backslash alpha)\}(x)\; =\; (n+\backslash alpha)!\; L\_n^\{(\backslash alpha)\}(x).$$

There is yet another, albeit less frequently used, convention in the physics literature {{cite book |last1=Schiff |first1=Leonard I. |title=Quantum mechanics |date=1968 |publisher=McGraw-Hill |location=New York |isbn=0070856435 |edition=3d}}{{cite book |last1=Messiah |first1=Albert |title=Quantum Mechanics. |date=2014 |publisher=Dover Publications |isbn=9780486784557}}{{cite book |last1=Boas |first1=Mary L. |title=Mathematical methods in the physical sciences |date=2006 |publisher=Wiley |location=Hoboken, NJ |isbn=9780471198260 |edition=3rd}}

$$\backslash tilde\{L\}\_n^\{(\backslash alpha)\}(x)\; =\; (-1)^\{\backslash alpha\}\backslash bar\{L\}\_\{n-\backslash alpha\}^\{(\backslash alpha)\}.$$

Generalized Laguerre polynomials are linked to Umbral calculus by being Sheffer sequences for $D/(D-I)$ when multiplied by $n!$. In Umbral Calculus convention,{{Cite journal |last1=Rota |first1=Gian-Carlo |last2=Kahaner |first2=D |last3=Odlyzko |first3=A |date=1973-06-01 |title=On the foundations of combinatorial theory. VIII. Finite operator calculus |journal=Journal of Mathematical Analysis and Applications |language=en |volume=42 |issue=3 |pages=684–760 |doi=10.1016/0022-247X(73)90172-8 |issn=0022-247X|doi-access=free }} the default Laguerre polynomials are defined to be$$\backslash mathcal\; L\_n(x)\; =\; n!L\_n^\{(-1)\}(x)\; =\; \backslash sum\_\{k=0\}^n\; L(n,k)\; (-x)^k$$where $L(n,k)\; =\; \backslash binom\{n-1\}\{k-1\}\; \backslash frac\{n!\}\{k!\}$ are the signless Lah numbers. $(\backslash mathcal\; L\_n(x))\_\{n\backslash in\backslash N\}$ is a sequence of polynomials of binomial type, ie they satisfy$$\backslash mathcal\; L\_n(x+y)\; =\; \backslash sum\_\{k=0\}^n\; \backslash binom\{n\}\{k\}\; \backslash mathcal\; L\_k(x)\; \backslash mathcal\; L\_\{n-k\}(y)$$

• Orthogonal polynomials
• Rodrigues' formula
• Angelescu polynomials
• Bessel polynomials
• Denisyuk polynomials
• Transverse mode, an important application of Laguerre polynomials to describe the field intensity within a waveguide or laser beam profile.

{{Reflist|35em}}

• {{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 }}

• {{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}}

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Category:Polynomials

Category:Orthogonal polynomials

Category:Special hypergeometric functions