Lagerra polynomials

Lagerra polynomials
Lagerra polynomials
general information
Formula
Scalar product
Domain
additional characteristics
Differential equation
Named after	Lagerr, Edmond Nicola

In mathematics, the Lagerr polynomials , named after Edmond Lagerr (1834–1886), are canonical solutions of the Lagerr equation :

x\,y''+(1-x)\,y'+n\,y=0,

{\ displaystyle x \, y '' + (1-x) \, y '+ n \, y = 0,}

x \, y '' + (1-x) \, y '+ n \, y = 0,

which is a linear differential equation of the second order. In physical kinetics, these same polynomials (sometimes accurate to normalization) are usually called Sonin or Sonin – Lagerra polynomials ^[1] . Laguerre polynomials are also used in the Gauss - Laguerre quadrature formula for the numerical calculation of integrals of the form:

\int \limits _{0}^{\infty }f(x)e^{-x}\,dx.

{\ displaystyle \ int \ limits _ {0} ^ {\ infty} f (x) e ^ {- x} \, dx.}

\ int \ limits_0 ^ \ infty f (x) e ^ {- x} \, dx.

Lagerra polynomials, usually denoted as $L_{0},\;L_{1},\;\ldots$ ${\ displaystyle L_ {0}, \; L_ {1}, \; \ ldots}$ $L_0, \; L_1, \; \ ldots$ are a sequence of polynomials that can be found by the Rodrigue formula

L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x}x^{n}\right)=\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}{n \choose k}x^{k}.

{\ displaystyle L_ {n} (x) = {\ frac {e ^ {x}} {n!}} {\ frac {d ^ {n}} {dx ^ {n}}} \ left (e ^ { -x} x ^ {n} \ right) = \ sum _ {k = 0} ^ {n} {\ frac {(-1) ^ {k}} {k!}} {n \ choose k} x ^ {k}.}

L_n (x) = \ frac {e ^ x} {n!} \ Frac {d ^ n} {dx ^ n} \ left (e ^ {- x} x ^ n \ right) = \ sum ^ {n} _ {k = 0} \ frac {(- 1) ^ k} {k!} {n \ choose k} x ^ k.

These polynomials are orthogonal to each other with a scalar product :

\langle f,\;g\rangle =\int \limits _{0}^{\infty }f(x)g(x)e^{-x}\,dx.

{\ displaystyle \ langle f, \; g \ rangle = \ int \ limits _ {0} ^ {\ infty} f (x) g (x) e ^ {- x} \, dx.}

\ langle f, \; g \ rangle = \ int \ limits_0 ^ \ infty f (x) g (x) e ^ {- x} \, dx.

The Laguerre polynomial sequence is a Schaeffer sequence .

Lagerra polynomials are used in quantum mechanics, in the radial part of the solution of the Schrödinger equation for an atom with one electron.

There are other applications of Lagerra polynomials.

Content

The first few polynomials

The following table lists the first few Lagerra polynomials:

$n$ ${\ displaystyle n}$	$L_{n}(x)$ ${\ displaystyle L_ {n} (x)}$
0	$one$ ${\ displaystyle 1}$
one	$-x+1$ ${\ displaystyle -x + 1}$
2	${\scriptstyle {\frac {1}{2}}}(x^{2}-4x+2)$ ${\ displaystyle {\ scriptstyle {\ frac {1} {2}}} (x ^ {2} -4x + 2)}$
3	${\scriptstyle {\frac {1}{6}}}(-x^{3}+9x^{2}-18x+6)$ ${\ displaystyle {\ scriptstyle {\ frac {1} {6}}} (- x ^ {3} + 9x ^ {2} -18x + 6)}$
four	${\scriptstyle {\frac {1}{24}}}(x^{4}-16x^{3}+72x^{2}-96x+24)$ ${\ displaystyle {\ scriptstyle {\ frac {1} {24}}} (x ^ {4} -16x ^ {3} + 72x ^ {2} -96x + 24)}$
five	${\scriptstyle {\frac {1}{120}}}(-x^{5}+25x^{4}-200x^{3}+600x^{2}-600x+120)$ ${\ displaystyle {\ scriptstyle {\ frac {1} {120}}} (- x ^ {5} + 25x ^ {4} -200x ^ {3} + 600x ^ {2} -600x + 120)}$
6	${\scriptstyle {\frac {1}{720}}}(x^{6}-36x^{5}+450x^{4}-2400x^{3}+5400x^{2}-4320x+720)$ ${\ displaystyle {\ scriptstyle {\ frac {1} {720}}} (x ^ {6} -36x ^ {5} + 450x ^ {4} -2400x ^ {3} + 5400x ^ {2} -4320x + 720)}$

The first 6 polynomials of Lagerra.

Recurrence Formula

Lagerra polynomials can be defined by the recurrence formula:

L_{k+1}(x)={\frac {1}{k+1}}{\bigl [}(2k+1-x)L_{k}(x)-kL_{k-1}(x){\bigr ]},\quad \forall k\geqslant 1,

{\ displaystyle L_ {k + 1} (x) = {\ frac {1} {k + 1}} {\ bigl [} (2k + 1-x) L_ {k} (x) -kL_ {k-1 } (x) {\ bigr]}, \ quad \ forall k \ geqslant 1,}

by predefining the first two polynomials as:

L_{0}(x)=1,

{\ displaystyle L_ {0} (x) = 1,}

L_{1}(x)=1-x.

{\ displaystyle L_ {1} (x) = 1-x.}

Generalized Lagerra polynomials

Generalized Lagerra polynomials $L_{n}^{a}(x)$ ${\ displaystyle L_ {n} ^ {a} (x)}$ are solutions of the equation:

x\,y''+(a+1-x)\,y'+n\,y=0,

{\ displaystyle x \, y '' + (a + 1-x) \, y '+ n \, y = 0,}

so that $L_{n}(x)=L_{n}^{0}(x)$ ${\ displaystyle L_ {n} (x) = L_ {n} ^ {0} (x)}$ .

Notes

↑ Lifshits, E.M. , Pitaevsky, L.P. Physical Kinetics. - ( Theoretical Physics , Volume X).

[1] Lifshits, E.M. , Pitaevsky, L.P. Physical Kinetics. - ( Theoretical Physics , Volume X).