FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2002, VOLUME 8, NUMBER 4, PAGES 1159-1178
V. F. Tarasov
Abstract
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Exact formulae for calculation of zeroes of Kummer's polynomials at
$a \le 4$
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
${}_1F_1(-4; c; z) = 0$
are based on \emph{one} (\emph{common} for all methods) equation of cubic
resolvent of FEL-type. For greater geometrical clarity of
(\emph{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.
All articles are published in Russian.
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Last modified: April 10, 2003