
On approximation of the values of some hypergeometric functions with irrational parameters
P. L. Ivankov^{} ^{} Bauman Moscow State Technical University
Abstract:
In this paper we consider some hypergeometric functions whose
parameters are connected in a special way. Lower estimates of the
moduli of linear forms in the values of such functions have been
obtained. Usually for the achievement of such estimates one makes
use of Siegel's method; see [1], [2],
[3, chapt. 3]. In this method the reasoning begins with the
construction by means of Dirichlet principle of the linear
approximating form having a sufficiently large order of zero at the
origin of coordinates. Employing the system of differential
equations, the functions under consideration satisfy, one constructs
then a set of forms such that the determinant composed of the
coefficients of the forms belonging to this set must not be equal to
zero identically. Further steps consist of constructing a set of
numerical forms and of proving of the interesting for the researcher
assertions: linear independence of the values of the functions under
consideration can be proved or corresponding quantitative results
can be obtained. By means of Siegel's method have been proved
sufficiently general theorems concerning the arithmetic nature of
the values of the generalized hypergeometric functions and in
addition to aforementioned linear independence in many cases was
established the transcendence and algebraic independence of the
values of such functions. But the employment of Dirichlet principle
at the first step of reasoning restricts the possibilities of the
method. Its direct employment is possible in the case of
hypergeometric functions with rational parameters only. It must be
taken into consideration also the insufficient accuracy of the
quantitative results that can be obtained by this method. As a
consequence of these facts some analogue of Siegel's method has been
developed (see [4]) by means of which it became possible in
some cases to investigate the arithmetic nature of the values of
hypergeometric functions with irrational parameters also.
But yet earlier one had begun to apply methods based on
effective construction of linear approximating form. By means of
such constructions the arithmetic nature of some classic constants
was investigated and corresponding quantitative results were
obtained, see for example [5, chapt. 1]. Subsequently it
turned out that effective methods can be applied also for the
investigation of generalized hypergeometric functions. Explicit
formulae for the coefficients of the linear approximating forms were
obtained. In some cases these formulae make it possible to realize
Siegel method scheme also for the hypergeometric functions with
irrational parameters. If in (1) polynomial $a(x)$ is equal
to unity identically then the results obtained by effective method
are of sufficiently general nature and in this case further
development of this method meets the obstacles of principal
character. In case $a(x)\not\equiv1$, however, the possibilities of
effective method are not yet exhausted and the latest results can be
generalized and improved.
In the theorems proved in the present paper new qualitative and
quantitative results are obtained for some hypergeometric functions
with $a(x)=x+\alpha$ and polynomial $b(x)$ from (1) of
special character. The case of irrational parameters is under
consideration but the ideas we use will apparently make it possible
in the future to obtain new results in case of rational parameters
also.
Bibliography: 15 titles.
Keywords:
generalized hypergeometric functions, irrational parameters, estimates of linear forms.
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UDC:
511.361 Received: 15.12.2015 Accepted:10.03.2016
Citation:
P. L. Ivankov, “On approximation of the values of some hypergeometric functions with irrational parameters”, Chebyshevskii Sb., 17:1 (2016), 108–116
Citation in format AMSBIB
\Bibitem{Iva16}
\by P.~L.~Ivankov
\paper On approximation of the values of some hypergeometric functions with irrational parameters
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 1
\pages 108116
\mathnet{http://mi.mathnet.ru/cheb456}
\elib{http://elibrary.ru/item.asp?id=25795073}
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