
On linear independence of the values of some hypergeometric functions over the imaginary quadratic field
P. L. Ivankov^{} ^{} Bauman Moscow
state technical University (Moscow)
Abstract:
The main difficulty one has to deal with while investigating
arithmetic nature of the values of the generalized hypergeometric
functions with irrational parameters consists in the fact that the
least common denominator of several first coefficients of the
corresponding power series increases too fast with the growth of
their number. The last circumstance makes it impossible to apply
known in the theory of transcendental numbers Siegel's method
for carrying out the above mentioned investigation. The application
of this method implies usage of pigeonhole principle for the
construction of a functional linear approximating form. This
construction is the first step in a long and complicated reasoning
that leads ultimately to the required arithmetic result. The
attempts to apply pigeonhole principle in case of functions with
irrational parameters encounters insurmountable obstacles because of
the aforementioned fast growth of the least common denominator of
the coefficients of the corresponding Taylor series. Owing to this
difficulty one usually applies effective construction of the linear
approximating form (or a system of such forms in case of
simultaneous approximations) for the functions with irrational
parameters. The effectively constructed form contains polynomials
with algebraic coefficients and it is necessary for further
reasoning to obtain a satisfactory upper estimate of the modulus of
the least common denominator of these coefficients. The known
estimates of this type should be in some cases improved. This
improvement is carried out by means of the theory of divisibility in
quadratic fields. Some facts concerning the distribution of the
prime numbers in arithmetic progression are also made use of. In the present paper we consider one of the versions of effective
construction of the simultaneous approximations for the
hypergeometric function of the general type and its derivatives. The
least common denominator of the coefficients of the polynomials
included in these approximations is estimated subsequently by means
of the improved variant of the corresponding lemma. All this makes
it possible to obtain a new result concerning the arithmetic values
of the aforesaid function at a nonzero point of small modulus from
some imaginary quadratic field.
Keywords:
hypergeometric function, effective construction, linear independence, imaginary quadratic field.
DOI:
https://doi.org/10.22405/222683832018203272281
Full text:
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UDC:
511.361 Received: 23.09.2019 Accepted:12.11.2019
Citation:
P. L. Ivankov, “On linear independence of the values of some hypergeometric functions over the imaginary quadratic field”, Chebyshevskii Sb., 20:3 (2019), 272–281
Citation in format AMSBIB
\Bibitem{Iva19}
\by P.~L.~Ivankov
\paper On linear independence of the values of some hypergeometric functions over the imaginary quadratic field
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 3
\pages 272281
\mathnet{http://mi.mathnet.ru/cheb811}
\crossref{https://doi.org/10.22405/222683832018203272281}
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