(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 4, PAGES 1159-1178

## Zeroes of Schrödinger's radial function $R$nl(r) and Kummer's function 1F1(-a;c;z) ($n < 10$, $l < 4$)

V. F. Tarasov

Abstract

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Exact formulae for calculation of zeroes of Kummer's polynomials at $a$£ 4 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 1F1(-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 = zk-(c+a-1) on the axis $y = 0$ the "circular" diagrams with the radius $R$a = (a-1)√(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 $2$£ c < ¥ for the cases $a = 3$ and $a = 4$ are obtained. While calculating zeroes $x$k of the $R$nl(r) and 1F1 functions, the "singular" cases $\left(a,c\right) = \left(4,6\right), \left(6,4\right), \left(8,14\right),$¼ are found.

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