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In mathematics, the Askey–Gasper inequality is an inequality for
Jacobi polynomials proved by
Richard Askey and
George Gasper (
1976 ) and used in the proof of the
Bieberbach conjecture .
It states that if
β
≥
0
{\displaystyle \beta \geq 0}
,
α
+
β
≥
−
2
{\displaystyle \alpha +\beta \geq -2}
, and
−
1
≤
x
≤
1
{\displaystyle -1\leq x\leq 1}
then
∑
k
=
0
n
P
k
(
α
,
β
)
(
x
)
P
k
(
β
,
α
)
(
1
)
≥
0
{\displaystyle \sum _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0}
where
P
k
(
α
,
β
)
(
x
)
{\displaystyle P_{k}^{(\alpha ,\beta )}(x)}
is a Jacobi polynomial.
The case when
β
=
0
{\displaystyle \beta =0}
can also be written as
3
F
2
(
−
n
,
n
+
α
+
2
,
1
2
(
α
+
1
)
;
1
2
(
α
+
3
)
,
α
+
1
;
t
)
>
0
,
0
≤
t
<
1
,
α
>
−
1.
{\displaystyle {}_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)>0,\qquad 0\leq t<1,\quad \alpha >-1.}
In this form, with α a non-negative integer, the inequality was used by
Louis de Branges in his proof of the
Bieberbach conjecture .
Ekhad (
1993 ) gave a short proof of this inequality, by combining the identity
(
α
+
2
)
n
n
!
×
3
F
2
(
−
n
,
n
+
α
+
2
,
1
2
(
α
+
1
)
;
1
2
(
α
+
3
)
,
α
+
1
;
t
)
=
=
(
1
2
)
j
(
α
2
+
1
)
n
−
j
(
α
2
+
3
2
)
n
−
2
j
(
α
+
1
)
n
−
2
j
j
!
(
α
2
+
3
2
)
n
−
j
(
α
2
+
1
2
)
n
−
2
j
(
n
−
2
j
)
!
×
3
F
2
(
−
n
+
2
j
,
n
−
2
j
+
α
+
1
,
1
2
(
α
+
1
)
;
1
2
(
α
+
2
)
,
α
+
1
;
t
)
{\displaystyle {\begin{aligned}{\frac {(\alpha +2)_{n}}{n!}}&\times {}_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)=\\&={\frac {\left({\tfrac {1}{2}}\right)_{j}\left({\tfrac {\alpha }{2}}+1\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-2j}(\alpha +1)_{n-2j}}{j!\left({\tfrac {\alpha }{2}}+{\tfrac {3}{2}}\right)_{n-j}\left({\tfrac {\alpha }{2}}+{\tfrac {1}{2}}\right)_{n-2j}(n-2j)!}}\times {}_{3}F_{2}\left(-n+2j,n-2j+\alpha +1,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +2),\alpha +1;t\right)\end{aligned}}}
with the
Clausen inequality .
Gasper & Rahman (2004 , 8.9) give some generalizations of the Askey–Gasper inequality to
basic hypergeometric series .
Askey, Richard ; Gasper, George (1976), "Positive Jacobi polynomial sums. II",
American Journal of Mathematics , 98 (3): 709–737,
doi :
10.2307/2373813 ,
ISSN
0002-9327 ,
JSTOR
2373813 ,
MR
0430358
Askey, Richard; Gasper, George (1986),
"Inequalities for polynomials" , in Baernstein, Albert; Drasin, David; Duren, Peter; Marden, Albert (eds.), The Bieberbach conjecture (West Lafayette, Ind., 1985) , Math. Surveys Monogr., vol. 21, Providence, R.I.:
American Mathematical Society , pp. 7–32,
ISBN
978-0-8218-1521-2 ,
MR
0875228
Ekhad, Shalosh B. (1993), Delest, M.; Jacob, G.; Leroux, P. (eds.), "A short, elementary, and easy, WZ proof of the Askey-Gasper inequality that was used by de Branges in his proof of the Bieberbach conjecture",
Theoretical Computer Science , Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991), 117 (1): 199–202,
doi :
10.1016/0304-3975(93)90313-I ,
ISSN
0304-3975 ,
MR
1235178
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series , Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.),
Cambridge University Press ,
ISBN
978-0-521-83357-8 ,
MR
2128719