Zeroes of Schr\"odinger's radial function $R_{nl}(r)$ and Kummer's function ${}_1F_{1}(-a;c;z)$ ($n10$, $l4$)

V. F. Tarasov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1159-1178

Résumé

Exact formulae for calculation of zeroes of Kummer's polynomials at $a\le4$ are given; in other cases ($a>4$) their numerical values (to within $10^{-15}$) are given. It is shown that the methods of L. Ferrari, L. Euler and J.-L. Lagrange that are used for solving the equation ${}_1F_1(-4;c;z)=0$ are based on one (common for all methods) equation of cubic resolvent of FEL-type. For greater geometrical clarity of (nonuniform for $a>3$) distribution of zeroes $x_{k}=z_{k}-(c+a-1)$ on the axis $y=0$ the “circular” diagrams with the radius $R_{a}=(a-1)\sqrt {c+a-1}$ are introduced for the first time. It allows to notice some singularities of distribution of these zeroes and their “images”, i. e. the points $T_{k}$ on the circle. Exact “angle” asymptotics of the points $T_{k}$ for $2\le c\infty$ for the cases $a=3$ and $a=4$ are obtained. While calculating zeroes $x_{k}$ of the $R_{nl}(r)$ and ${}_1F_1$ functions, the “singular” cases $(a,c)=(4,6),(6,4),(8,14),\ldots$ are found.