In mathematics, the Askey–Gasper inequality is an inequality for

Jacobi polynomial In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The ...

s proved by and used in the proof of the

Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It was ...

Statement

It states that if

\beta\geq 0

\alpha+\beta\geq -2

, and

-1\leq x\leq 1

then :

\sum_^n \frac \ge 0

where :

P_k^(x)

is a Jacobi polynomial. The case when

\beta=0

can also be written as :

_3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right)>0, \qquad 0\leq t<1, \quad \alpha>-1.

In this form, with a non-negative integer, the inequality was used by

Louis de Branges Louis may refer to: People * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer Other uses * Louis (coin), a French coin * HMS ''Louis'', two ships of the Royal Navy See also ...

in his proof of the

Proof

gave a short proof of this inequality, by combining the identity :

\begin
&\frac\times _3F_2 \left (-n,n+\alpha+2,\tfrac(\alpha+1);\tfrac(\alpha+3),\alpha+1;t \right)\\
=&\sum_ \frac \times _3F_2\left (-n+2j,n-2j+\alpha+1,\tfrac(\alpha+1);\tfrac(\alpha+2),\alpha+1;t \right )
\end

with the

Clausen inequality Clausen is a Danish language, Danish patronymic surname, literally meaning ''child of Claus'', Claus being a German language, German form of the Greek language, Greek Νικόλαος, Nikolaos, (cf. Nicholas), used in Denmark at least since the 16t ...

Generalizations

give some generalizations of the Askey–Gasper inequality to

basic hypergeometric series In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are q-analog, ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is ...

References

* * * * {{DEFAULTSORT:Askey-Gasper inequality Inequalities (mathematics) Special functions Orthogonal polynomials