
On differentiation with respect to parameter
P. L. Ivankov^{} ^{} Bauman Moscow State Technical University
Abstract:
The investigation of the arithmetic nature of the values of
differentiated with respect to parameter generalized hypergeometric
functions was carried out in many works; see [1]–[7]
and also corresponding chapters of the books [8] and
[9]. Primarily the method of Siegel was used for these
purposes. This method can be applied for the investigation of
hypergeometric functions with rational parameters and the results
concerning transcendence and algebraic independence of the values of
such functions and corresponding quantitative results (for example
estimates of the measures of algebraic independence) were obtained
by means of it. The possibilities of application of Siegel's method
in case of hypergeometric functions with irrational parameters are
restricted. In its classic form Siegel's method cannot be applied in
this situation and here were required some new considerations. But
it must be noted that the most general results concerning the
arithmetic nature of the values of hypergeometric functions with
irrational parameters were obtained exactly by Siegel's method (by
modified form of it, see [10] and [11]). In this case
it's impossible to say of the results of transcendence or algebraic
independence and one must restrict oneself by the results concerning
linear independence of the corresponding values.
In Siegel's method reasoning begins with the construction of
functional linear approximating form which has a sufficiently high
order of zero at the origin of coordinates. Such a form is
constructed by means of the Dirichlet principle. The impossibility
to realize the corresponding reasoning for the functions with
irrational parameters is an obstacle for the attempts to apply
Siegel's method in case of irrational parameters.
It was noted long ago that in some cases the linear approximating
form can be constructed effectively and explicit formulae can be
pointed out for its coefficients. This method is inferior to
Siegel's one in the sense of the generality of the results obtained.
But by means of the method based on the effective construction of
linear approximating form the most precise low estimates of the
modules of linear forms in the values of hypergeometric functions
were obtained and in many cases were established linear
independence of the values of functions with irrational parameters
(see for example [12]).
The effective construction of linear approximating form for the
function (2) was proposed in the work [13]. In this
work the construction was based on a contour integral which was
earlier used for the achievement of results concerning the estimates
of linear forms of the values of hypergeometric functions with
different parameters; see [14]. In this paper we propose a new
approach for the construction of linear approximating form for
functions (2). Here we make use of a connection between
hypergeometric functions of different types which makes it possible
to reduce above mentioned constructing of linear approximating form
to less difficult task. In the conclusion we give short directions
concerning possible applications.
Bibliography: 15 titles.
Keywords:
the simplest hypergeometric function, 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 differentiation with respect to parameter”, Chebyshevskii Sb., 16:3 (2015), 285–294
Citation in format AMSBIB
\Bibitem{Iva15}
\by P.~L.~Ivankov
\paper On differentiation with respect to parameter
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 3
\pages 285294
\mathnet{http://mi.mathnet.ru/cheb419}
\elib{http://elibrary.ru/item.asp?id=24398938}
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