Rodrigezova formula i polinomi
uredi
Lagerovi polinomi obično se označavaju kao L0 , 1 , ..., a polinomni niz može da se definiše Rodrigezovom formulom:
L
n
(
x
)
=
e
x
n
!
d
n
d
x
n
(
e
−
x
x
n
)
.
{\displaystyle L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x}x^{n}\right).}
Prvih nekoliko polinoma:
n
L
n
(
x
)
{\displaystyle L_{n}(x)\,}
0
1
{\displaystyle 1\,}
1
−
x
+
1
{\displaystyle -x+1\,}
2
1
2
(
x
2
−
4
x
+
2
)
{\displaystyle {\scriptstyle {\frac {1}{2}}}(x^{2}-4x+2)\,}
3
1
6
(
−
x
3
+
9
x
2
−
18
x
+
6
)
{\displaystyle {\scriptstyle {\frac {1}{6}}}(-x^{3}+9x^{2}-18x+6)\,}
4
1
24
(
x
4
−
16
x
3
+
72
x
2
−
96
x
+
24
)
{\displaystyle {\scriptstyle {\frac {1}{24}}}(x^{4}-16x^{3}+72x^{2}-96x+24)\,}
5
1
120
(
−
x
5
+
25
x
4
−
200
x
3
+
600
x
2
−
600
x
+
120
)
{\displaystyle {\scriptstyle {\frac {1}{120}}}(-x^{5}+25x^{4}-200x^{3}+600x^{2}-600x+120)\,}
6
1
720
(
x
6
−
36
x
5
+
450
x
4
−
2400
x
3
+
5400
x
2
−
4320
x
+
720
)
{\displaystyle {\scriptstyle {\frac {1}{720}}}(x^{6}-36x^{5}+450x^{4}-2400x^{3}+5400x^{2}-4320x+720)\,}
Generirajuća funkcija Lagerovih polinoma je:
e
−
x
t
/
(
1
−
t
)
1
−
t
=
∑
n
=
0
∞
L
n
(
x
)
t
n
{\displaystyle {\frac {e^{-xt/(1-t)}}{1-t}}=\sum _{n=0}^{\infty }L_{n}(x)t^{n}\,}
Rekurzivne relacije
uredi
Generalizirani Lagerovi polinomi
uredi
Generalizirani Lagerovi polinomi ili pridruženi Lagerovi polinomi
L
n
(
α
)
(
x
)
{\displaystyle L_{n}^{(\alpha )}(x)}
x
y
″
+
(
α
+
1
−
x
)
y
′
+
n
y
=
0
{\displaystyle x\,y''+(\alpha +1-x)\,y'+n\,y=0}
Rodrigezova formula za generalizirane polinome je:
L
n
(
α
)
(
x
)
=
x
−
α
e
x
n
!
d
n
d
x
n
(
e
−
x
x
n
+
α
)
.
{\displaystyle L_{n}^{(\alpha )}(x)={x^{-\alpha }e^{x} \over n!}{d^{n} \over dx^{n}}\left(e^{-x}x^{n+\alpha }\right).}
Veza običnih i generaliziranih Lagerovih polinoma je:
L
n
k
(
x
)
=
(
−
1
)
k
d
k
d
x
k
L
n
+
k
(
x
)
{\displaystyle L_{n}^{k}(x)=(-1)^{k}\,{\frac {{\rm {d}}^{k}}{{\rm {d}}x^{k}}}\,L_{n+k}(x)}
Obični Lagerovi polinomi ekvivalentni su generaliziranima polinomima ako je α = 0:
L
n
(
0
)
(
x
)
=
L
n
(
x
)
.
{\displaystyle L_{n}^{(0)}(x)=L_{n}(x).}
Nekoliko prvih generaliziranih Legerovih polinoma:
L
0
k
(
x
)
=
1
{\displaystyle L_{0}^{k}(x)=1}
L
1
k
(
x
)
=
−
x
+
k
+
1
{\displaystyle L_{1}^{k}(x)=-x+k+1}
L
2
k
(
x
)
=
1
2
[
x
2
−
2
(
k
+
2
)
x
+
(
k
+
1
)
(
k
+
2
)
]
{\displaystyle L_{2}^{k}(x)={\frac {1}{2}}\,\left[x^{2}-2\,(k+2)\,x+(k+1)(k+2)\right]}
L
3
k
(
x
)
=
1
6
[
−
x
3
+
3
(
k
+
3
)
x
2
−
3
(
k
+
2
)
(
k
+
3
)
x
+
(
k
+
1
)
(
k
+
2
)
(
k
+
3
)
]
{\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]}
Ortogonalnost
uredi
Pridruženi Lagerovi polinomi ortogonalni su u odnosu na težinsku funkciju
x
α
e
−
x
{\displaystyle x^{\alpha }e^{-x}}
∫
0
∞
x
α
e
−
x
L
n
(
α
)
(
x
)
L
m
(
α
)
(
x
)
d
x
=
Γ
(
n
+
α
+
1
)
n
!
δ
n
,
m
,
{\displaystyle \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},}
Veza sa Ermitovim polinomima
uredi
Generalizirani lagerovi polinomi povezani su sa Ermitovim polinomima sledećim relacijama:
H
2
n
(
x
)
=
(
−
1
)
n
2
2
n
n
!
L
n
(
−
1
/
2
)
(
x
2
)
{\displaystyle H_{2n}(x)=(-1)^{n}2^{2n}n!L_{n}^{(-1/2)}(x^{2})}
i
H
2
n
+
1
(
x
)
=
(
−
1
)
n
2
2
n
+
1
n
!
x
L
n
(
1
/
2
)
(
x
2
)
{\displaystyle H_{2n+1}(x)=(-1)^{n}2^{2n+1}n!xL_{n}^{(1/2)}(x^{2})}
gde su
H
n
(
x
)
{\displaystyle H_{n}(x)}
Ermitovi polinomi .
Literatura
uredi
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 978-0486612720
![{\displaystyle 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)
