Geodesic
Zeroes of Schrödinger's radial function $R_{nl}(r)$ and Kummer's function ${}_1F_{1}(-a;c;z)$ ($n10$, $l4$)
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1159-1178 
V. F. Tarasov. Zeroes of Schrödinger's radial function $R_{nl}(r)$ and Kummer's function ${}_1F_{1}(-a;c;z)$ ($n<10$, $l<4$). Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 1159-1178. http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a14/
@article{FPM_2002_8_4_a14,
     author = {V. F. Tarasov},
     title = {Zeroes of {Schr\"odinger's} radial function $R_{nl}(r)$ and {Kummer's} function ${}_1F_{1}(-a;c;z)$ ($n<10$, $l<4$)},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1159--1178},
     year = {2002},
     volume = {8},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a14/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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