As for the \({}_2F_3\) hypergeometric function of the form

$$\begin{aligned} {}_2F_3\left[ \begin{array}{c} a_1, a_2\\ b_1, b_2, b_3\end{array}\biggr | -x^2\right] \qquad (x>0), \end{aligned}$$

where all of parameters are assumed to be positive, we give sufficient conditions on \((b_1, b_2, b_3)\) for its positivity in terms of Newton polyhedra with vertices consisting of permutations of \(\,(a_2, a_1+1/2, 2a_1)\,\) or \(\,(a_1, a_2+1/2, 2a_2).\) As an application, we obtain an extensive validity region of \((\alpha , \lambda , \mu )\) for the inequality

$$\begin{aligned} \int _0^x (x-t)^{\lambda }\, t^{\mu } J_\alpha (t)\, dt \ge 0\qquad (x>0). \end{aligned}$$

中文翻译：

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牛顿多面体和$$ {} _ 2F_3 $$ 2 F 3超几何函数的正性

至于形式的\（{} _ 2F_3 \）超几何函数

$$ \ begin {aligned} {} _2F_3 \ left [\ begin {array} {c} a_1，a_2 \\ b_1，b_2，b_3 \ end {array} \ biggr | -x ^ 2 \ right] \ qquad（x> 0），\ end {aligned} $$

在所有参数均假定为正的情况下，我们对\（（b_1，b_2，b_3）\）的正定性给出了充分的条件，其条件为牛顿多面体，其顶点包含\（\，（a_2，a_1 + 1）的排列/ 2，2a_1）\，\）或\（\，（a_1，a_2 + 1/2，2a_2）。\）作为应用程序，我们获得了\（（\ alpha，\ lambda，\ mu ）\）用于不等式

$$ \ begin {aligned} \ int _0 ^ x（xt）^ {\ lambda} \，t ^ {\ mu} J_ \ alpha（t）\，dt \ ge 0 \ qquad（x> 0）。\ end {aligned} $$