Askey–Gasper inequality

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

  • 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