
On simultaneous approximations
P. L. Ivankov^{} ^{} Bauman Moscow State Technical University
Abstract:
In this paper we consider hypergeometric functions and their
derivatives (see (2) and (3)). One begins the
investigation of arithmetic nature of the values of such functions
with the construction of functional linear approximating form
having sufficiently high order of zero at the origin. If the
parameters of functions under consideration (in our case these are
numbers (1)) are rational the construction of such a form can
be fulfilled by means of the Dirichlet principle. Further reasoning
is based on the employment of the constructed form and the whole
scheme is called Siegel's method, see [1] and [2]. If
some of the numbers (1) are irrational the functions
(2) and (3) cannot be reduced to the so called
$E$functions and it is impossible to use Siegel's method (in its
classic form) for such functions: the scheme doesn't work at the
very beginning of reasoning for we cannot use the Dirichlet
principle for the construction of the first approximating linear
functional form (in the process of reasoning by Siegel's method we
get several such forms). It was noticed that in some cases the first
approximating form can be constructed effectively (see for example [3] and [4]). Having at one's disposal such a form one
can reason as in Siegel's method (or it is possible in some cases to
use special properties of the effectively constructed linear form)
and receive required results. These results are not so general as
those received by Siegel's method but the method based on
effectively constructed approximating form has its own advantages.
One of them consists in the possibility of its application also in
case when some of the parameters (1) are irrational. The
other advantage is the more precise estimates (if we consider for
instance the measure of linear independence) that can be obtained by
this method.
The above concerns the case when the functions under consideration
are not differentiated with respect to parameter. Application of
Siegel's method for the differentiated with respect to parameter
functions (for example such functions as (4) and (5))
is possible also and it has been in fact fulfilled in a series of
works; see the remarks to chapter 7 of the book by A. B. Shidlovskii [5]. But as before the parameters of the functions under
consideration must be rational and the obtained results are not
sufficiently precise.
The performed investigations show that the employment of
simultaneous approximations instead of construction of linear
approximating form almost always gives better results. For that
reason the main new results concerning differentiated with respect
to parameter hypergeometric functions have been obtained exactly by
means of the effective constructions of simultaneous
approximations although the appearance (comparatively recently)
of effective constructions of linear approximating forms for
such functions did make it possible to solve some related problems.
In this paper we propose a new effective construction of
simultaneous approximations for the differentiated with respect to
parameter hypergeometric functions in homogeneous case. On
possible applications of this construction we give only brief
instructions: one can obtain some results on linear independence of
the values of functions of the type (5) in case of
irrationality of some of the numbers (1); it is possible also
to improve some of the related quantitative results.
Bibliography: 15 titles.
Keywords:
generalized hypergeometric functions, differentiation with respect to parameter, estimates of linear forms.
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UDC:
511.361 Received: 31.05.2015
Citation:
P. L. Ivankov, “On simultaneous approximations”, Chebyshevskii Sb., 16:3 (2015), 295–305
Citation in format AMSBIB
\Bibitem{Iva15}
\by P.~L.~Ivankov
\paper On simultaneous approximations
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 3
\pages 295305
\mathnet{http://mi.mathnet.ru/cheb420}
\elib{http://elibrary.ru/item.asp?id=24398939}
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