戈特利布多項式 是一個以超幾何函數 定義的正交多項式
ℓ
n
(
x
,
λ
)
=
e
−
n
λ
∑
k
(
1
−
e
λ
)
k
(
n
k
)
(
x
k
)
=
e
−
n
λ
2
F
1
(
−
n
,
−
x
;
1
;
1
−
e
λ
)
{\displaystyle \displaystyle \ell _{n}(x,\lambda )=e^{-n\lambda }\sum _{k}(1-e^{\lambda })^{k}{\binom {n}{k}}{\binom {x}{k}}=e^{-n\lambda }{}_{2}F_{1}(-n,-x;1;1-e^{\lambda })}
Gottlieb Polynomials
ℓ
0
(
x
,
λ
)
=
1
{\displaystyle \displaystyle \ell _{0}(x,\lambda )=1}
ℓ
1
(
x
,
λ
)
=
−
e
x
p
(
−
λ
)
∗
(
−
1
−
x
+
x
∗
e
x
p
(
λ
)
)
{\displaystyle \displaystyle \ell _{1}(x,\lambda )=-exp(-\lambda )*(-1-x+x*exp(\lambda ))}
ℓ
2
(
x
,
λ
)
=
−
(
1
/
2
)
∗
e
x
p
(
−
2
∗
λ
)
∗
(
−
2
−
3
∗
x
+
2
∗
x
∗
e
x
p
(
λ
)
−
x
2
+
2
∗
x
2
∗
e
x
p
(
λ
)
−
e
x
p
(
2
∗
λ
)
∗
x
2
+
e
x
p
(
2
∗
λ
)
∗
x
)
{\displaystyle \displaystyle \ell _{2}(x,\lambda )=-(1/2)*exp(-2*\lambda )*(-2-3*x+2*x*exp(\lambda )-x^{2}+2*x^{2}*exp(\lambda )-exp(2*\lambda )*x^{2}+exp(2*\lambda )*x)}
ℓ
3
(
x
,
λ
)
=
−
(
1
/
6
)
∗
e
x
p
(
−
3
∗
λ
)
∗
(
−
6
−
11
∗
x
+
6
∗
x
∗
e
x
p
(
λ
)
−
6
∗
x
2
+
9
∗
x
2
∗
e
x
p
(
λ
)
+
3
∗
e
x
p
(
2
∗
λ
)
∗
x
−
x
3
+
3
∗
x
3
∗
e
x
p
(
λ
)
−
3
∗
e
x
p
(
2
∗
λ
)
∗
x
3
+
e
x
p
(
3
∗
λ
)
∗
x
3
−
3
∗
e
x
p
(
3
∗
λ
)
∗
x
2
+
2
∗
e
x
p
(
3
∗
λ
)
∗
x
)
{\displaystyle \displaystyle \ell _{3}(x,\lambda )=-(1/6)*exp(-3*\lambda )*(-6-11*x+6*x*exp(\lambda )-6*x^{2}+9*x^{2}*exp(\lambda )+3*exp(2*\lambda )*x-x^{3}+3*x^{3}*exp(\lambda )-3*exp(2*\lambda )*x^{3}+exp(3*\lambda )*x^{3}-3*exp(3*\lambda )*x^{2}+2*exp(3*\lambda )*x)}
ℓ
4
(
x
,
λ
)
=
−
(
1
/
24
)
∗
e
x
p
(
−
4
∗
λ
)
∗
(
−
24
−
50
∗
x
+
24
∗
x
∗
e
x
p
(
λ
)
−
35
∗
x
2
−
e
x
p
(
4
∗
λ
)
∗
x
4
+
4
∗
x
4
∗
e
x
p
(
λ
)
−
6
∗
e
x
p
(
2
∗
λ
)
∗
x
4
+
4
∗
e
x
p
(
3
∗
λ
)
∗
x
4
+
6
∗
e
x
p
(
4
∗
λ
)
∗
x
−
11
∗
e
x
p
(
4
∗
λ
)
∗
x
2
+
6
∗
e
x
p
(
4
∗
λ
)
∗
x
3
+
8
∗
e
x
p
(
3
∗
λ
)
∗
x
−
4
∗
e
x
p
(
3
∗
λ
)
∗
x
2
+
24
∗
x
3
∗
e
x
p
(
λ
)
−
12
∗
e
x
p
(
2
∗
λ
)
∗
x
3
−
8
∗
e
x
p
(
3
∗
λ
)
∗
x
3
+
44
∗
x
2
∗
e
x
p
(
λ
)
+
6
∗
e
x
p
(
2
∗
λ
)
∗
x
2
+
12
∗
e
x
p
(
2
∗
λ
)
∗
x
−
10
∗
x
3
−
x
4
)
{\displaystyle \displaystyle \ell _{4}(x,\lambda )=-(1/24)*exp(-4*\lambda )*(-24-50*x+24*x*exp(\lambda )-35*x^{2}-exp(4*\lambda )*x^{4}+4*x^{4}*exp(\lambda )-6*exp(2*\lambda )*x^{4}+4*exp(3*\lambda )*x^{4}+6*exp(4*\lambda )*x-11*exp(4*\lambda )*x^{2}+6*exp(4*\lambda )*x^{3}+8*exp(3*\lambda )*x-4*exp(3*\lambda )*x^{2}+24*x^{3}*exp(\lambda )-12*exp(2*\lambda )*x^{3}-8*exp(3*\lambda )*x^{3}+44*x^{2}*exp(\lambda )+6*exp(2*\lambda )*x^{2}+12*exp(2*\lambda )*x-10*x^{3}-x^{4})}
Gottlieb, M. J., Concerning some polynomials orthogonal on a finite or enumerable set of points., American Journal of Mathematics , 1938, 60 : 453–458, ISSN 0002-9327 JFM 64.0329.01 doi:10.2307/2371307 Rainville, Earl D. , Special functions, New York: The Macmillan Co., 1960, MR 0107725 
