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Askey–Gasper inequality

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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.

Statement

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.

Proof

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.

Generalizations

Gasper & Rahman (2004, 8.9) give some generalizations of the Askey–Gasper inequality to basic hypergeometric series.

See also

References

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