Mathematics of the USSR-Sbornik
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mathematics of the USSR-Sbornik, 1970, Volume 11, Issue 3, Pages 355–376
DOI: https://doi.org/10.1070/SM1970v011n03ABEH002073
(Mi sm3457)
 

This article is cited in 3 scientific papers (total in 3 papers)

Transcendence and algebraic independence of the values of some hypergeometric $E$-functions

I. I. Belogrivov
References:
Abstract: We investigate the arithmetical character of the values of the functions
\begin{gather*} A_{m,s}(z)=\sum_{n=0}^\infty[\lambda_1+1,n]^{-m_1}[\lambda_2+1,n]^{-m_2}\cdots[\lambda_s+1,n]^{-m_s}\biggl(\frac zm\biggr)^{mn},\\ \lambda_1,\lambda_2,\dots,\lambda_s\ne-1,-2,\dots,\\ A_{m,s,\mu}(z)=1+\sum_{n=1}^\infty[\lambda_1+1,n]^{m_1}\cdots[\lambda_{i-1}+1,n]^{-m_{i-1}}\cdots[\lambda_i+1,n-1]^{-m_i}\cdots\\ \cdots[\lambda_s+1,n-1]^{-m_s}[\lambda_i+n]^{q^{i-1}-\mu}\biggl(\frac zm\biggr)^{mn}, \end{gather*}
$\lambda_1,\lambda_2,\dots,\lambda_s\ne-1,-2,\dots$; $\mu=q_{i-1}+1$, $q_{i-1}+2,\dots,q_i$, $i=1,2,\dots,s,$ where $s\geqslant1$; $\lambda_1,\lambda_2,\dots,\lambda_s$ are rational numbers; $[\lambda,0]=1$, $[\lambda,n]=\lambda(\lambda+1)\cdots(\lambda+n-1)$, $n\geqslant1$, $m_1,m_2,\dots,m_s$ are arbitrary nonnegative rational integers, $m_0=0$, $m=m_1+m_2+\dots+m_s$, $m\geqslant1$; $q_i=m_1+m_2+\dots+m_i$, $i=1,2,\dots,s-1$, $q_0=0$, $q=q_s=m_1+m_2+\dots+m_{s-1}+t^s$, $t_s\geqslant m_s$, $t_s$ a natural number.
The function $A_{m,s}(z)$ is the solution of a linear differential equation of order $m$ with polynomial coefficients. The system of functions $A_{m,s,\mu}(z)$, $\mu=1,2,\dots,q$, constitutes the solution of a system of $q$ linear differential equations whose coefficients are rational functions of $z$.
By means of the general theorem of Shidlovskii on the transcendence and algebraic independence of the values of the $E$-functions we prove six theorems on the mutual transcendence of the values of the functions in each aggregate $A_{m,s}(z), A'_{m,s}(z),\dots,A^{(m-1)}_{m,s}(z)$ and $A_{m,s,\mu}(z)$, $\mu=1,2,\dots,q$, at arbitrary algebraic points $a\ne0$ for various rational values of the parameters $\lambda_1,\lambda_2,\dots,\lambda_s$, $s\geqslant1$, and arbitrary values $m_1,m_2,\dots,m_s$.
Bibliography: 8 titles.
Received: 06.07.1969
Bibliographic databases:
UDC: 517.516
MSC: 33C60, 34K06
Language: English
Original paper language: Russian
Citation: I. I. Belogrivov, “Transcendence and algebraic independence of the values of some hypergeometric $E$-functions”, Math. USSR-Sb., 11:3 (1970), 355–376
Citation in format AMSBIB
\Bibitem{Bel70}
\by I.~I.~Belogrivov
\paper Transcendence and algebraic independence of the values of some hypergeometric $E$-functions
\jour Math. USSR-Sb.
\yr 1970
\vol 11
\issue 3
\pages 355--376
\mathnet{http://mi.mathnet.ru//eng/sm3457}
\crossref{https://doi.org/10.1070/SM1970v011n03ABEH002073}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=277483}
\zmath{https://zbmath.org/?q=an:0225.10036}
Linking options:
  • https://www.mathnet.ru/eng/sm3457
  • https://doi.org/10.1070/SM1970v011n03ABEH002073
  • https://www.mathnet.ru/eng/sm/v124/i3/p387
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:266
    Russian version PDF:89
    English version PDF:17
    References:38
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024