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 $β \geq 0, α + β \geq -2$ , and $-1 \leq x \leq 1$ then

\sum_{k=0}^n \frac{P_k^{(\alpha,\beta)}(x)}{P_k^{(\beta,\alpha)}(1)} \ge 0

where

P_k^{(\alpha,\beta)}(x)

is a Jacobi polynomial.

The case when $β = 0$ can also be written as

{}_3F_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

\begin{align} \frac{(\alpha+2)_n}{n!} &\times {}_3F_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 {}_3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac{1}{2}(\alpha+1);\tfrac{1}{2}(\alpha+2),\alpha+1;t \right ) \end{align}

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, American Journal of Mathematics, Vol. 98, No. 3, 98 (3): 709–737, ISSN 0002-9327, JSTOR 2373813, MR 0430358, doi:10.2307/2373813
Askey, Richard; Gasper, George (1986), "Inequalities for polynomials", in Baernstein, Albert; Drasin, David; Duren, Peter; Marden, Albert, 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, MR 875228
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, ISSN 0304-3975, MR 1235178, doi:10.1016/0304-3975(93)90313-I
Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574 (inactive 2017-01-31)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.