# アスキー＝ギャスパー不等式

## 主張

もしβ ≥ 0, α + β ≥ −2かつ−1 ≤ x ≤ 1 ならば

${\displaystyle \sum _{k=0}^{n}{\frac {P_{k}^{(\alpha ,\beta )}(x)}{P_{k}^{(\beta ,\alpha )}(1)}}\geq 0}$

であり、

${\displaystyle P_{k}^{(\alpha ,\beta )}(x)}$

はヤコビ多項式である。 β = 0のとき、

${\displaystyle {}_{3}F_{2}\left(-n,n+\alpha +2,{\tfrac {1}{2}}(\alpha +1);{\tfrac {1}{2}}(\alpha +3),\alpha +1;t\right)>0,\qquad 0\leq t<1,\quad \alpha >-1.}$

になる。

## 参考文献

• 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, MR0430358
• Askey, Richard; Gasper, George (1986), “Inequalities for polynomials”, in Baernstein, Albert; Drasin, David; Duren, Peter et al., The Bieberbach conjecture (West Lafayette, Ind., 1985), Math. Surveys Monogr., 21, Providence, R.I.: American Mathematical Society, pp. 7–32, ISBN 978-0-8218-1521-2, MR875228
• 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, MR1235178
• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR2128719