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Ojlerovi polinomi u matematici predstavljaju polinome, koji su dobili ime prema Leonardu Ojleru , a susreću se prilikom izučavanja mnogih specijalnih funkcija, a posebno Rimanove zeta funkcije i Hurvicove zeta funkcije . Blisko su povezani sa Bernulijevim polinomima .
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{\displaystyle E_{m}(x)=\sum _{n=0}^{m}{\frac {1}{2^{n}}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(x+k)^{m}}
,
gde su
(
n
k
)
{\displaystyle {n \choose k}}
— binomni koeficijenti
Generirajuća funkcija i članovi [ uredi | uredi izvor ]
Generirajuća funkcija Ojlerovih polinoma je:
2
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x
t
e
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1
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∞
E
n
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n
n
!
.
{\displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }E_{n}(x){\frac {t^{n}}{n!}}.}
Nekoliko prvih Ojlerovih polinoma:
E
0
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1
{\displaystyle E_{0}(x)=1}
E
1
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1
/
2
{\displaystyle E_{1}(x)=x-1/2}
E
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x
2
−
x
{\displaystyle E_{2}(x)=x^{2}-x}
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3
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3
2
x
2
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4
{\displaystyle E_{3}(x)=x^{3}-{\frac {3}{2}}x^{2}+{\frac {1}{4}}}
E
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x
4
−
2
x
3
+
x
{\displaystyle E_{4}(x)=x^{4}-2x^{3}+x}
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5
−
5
2
x
4
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2
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2
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{\displaystyle E_{5}(x)=x^{5}-{\frac {5}{2}}x^{4}+{\frac {5}{2}}x^{2}-{\frac {1}{2}}}
E
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x
6
−
3
x
5
+
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x
3
−
3
x
.
{\displaystyle E_{6}(x)=x^{6}-3x^{5}+5x^{3}-3x.}
E
n
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1
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+
E
n
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2
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n
{\displaystyle E_{n}(x+1)+E_{n}(x)=2x^{n}\,}
E
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n
E
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1
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{\displaystyle E_{n}'(x)=nE_{n-1}(x)\,}
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k
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E
k
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y
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{\displaystyle E_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}E_{k}(x)y^{n-k}\,}
E
n
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E
n
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{\displaystyle E_{n}(1-x)=(-1)^{n}E_{n}(x)\,}
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E
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{\displaystyle (-1)^{n}E_{n}(-x)=-E_{n}(x)+2x^{n}\,}
∫
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{\displaystyle \int _{a}^{x}E_{n}(t)\,dt={\frac {E_{n+1}(x)-E_{n+1}(a)}{n+1}}}
∫
0
1
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!
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!
B
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+
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{\displaystyle \int _{0}^{1}E_{n}(t)E_{m}(t)\,dt=(-1)^{n}4(2^{m+n+2}-1){\frac {m!n!}{(m+n+2)!}}B_{n+m+2}}
gde su
B
k
{\displaystyle \ B_{k}}
— Bernulijevi brojevi