Geodesic
Transcendence and algebraic independence of the values of some hypergeometric $E$-functions
Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 355-376 
I. I. Belogrivov. Transcendence and algebraic independence of the values of some hypergeometric $E$-functions. Sbornik. Mathematics, Tome 11 (1970) no. 3, pp. 355-376. http://geodesic.mathdoc.fr/item/SM_1970_11_3_a4/
@article{SM_1970_11_3_a4,
     author = {I. I. Belogrivov},
     title = {Transcendence and algebraic independence of the values of some hypergeometric $E$-functions},
     journal = {Sbornik. Mathematics},
     pages = {355--376},
     year = {1970},
     volume = {11},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_11_3_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] A. B. Shidlovskii, “O kriterii algebraicheskoi nezavisimosti znachenii odnogo klassa tselykh funktsii”, Izv. AN SSSR, seriya matem., 23 (1959), 35–66 | Zbl

[2] A. B. Shidlovskii, “O transtsendentnosti i algebraicheskoi nezavisimosti znachenii $E$-funktsii, svyazannykh lyubym chislom algebraicheskikh uravnenii v pole ratsionalnykh funktsii”, Izv. AN SSSR, seriya matem., 26 (1962), 877–910 | Zbl

[3] S. Siegel, “Über einige Anwendungen Diophantische Approximationen”, Abhandl. Pruss. Acad. Wiss., 1929–30, no. 1, 1–70 | Zbl

[4] A. B. Shidlovskii, “O transtsendentnosti i algebraicheskoi nezavisimosti znachenii nekotorykh $E$-funktsii”, Vestnik MGU, 1961, no. 5, 44–59

[5] A. B. Shidlovskii, “Ob algebraicheskoi nezavisimosti znachenii nekotorykh gipergeometricheskikh $E$-funktsii”, Trudy Mosk. matem. ob-va, 18 (1968), 55–64 | MR

[6] A. B. Shidlovskii, “O transtsendentnosti i algebraicheskoi nezavisimosti znachenii tselykh funktsii nekotorykh klassov”, DAN SSSR, 96:4 (1954), 697–699; Математика, т. IX, Ученые записки МГУ, 186, 1959, 11–70 | MR | Zbl

[7] V. A. Oleinikov, “O transtsendentnosti i algebraicheskoi nezavisimosti znachenii $E$-funktsii, yavlyayuschikhsya resheniem lineinogo differentsialnogo uravneniya tretego poryadka”, DAN SSSR, 166:3 (1966), 540–543 | Zbl

[8] A. B. Shidlovskii, “Ob otsenkakh mery transtsendentnosti znachenii $E$-funktsii”, Matem. zametki, 2:1 (1967), 33–34