Geodesic
Algebraic Relations between the Hypergeometric E-Function and Its Derivatives
Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 832-844.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider the generalized hypergeometric function $$ \sum _{n=0}^\infty \frac 1{(\lambda _1+1)_n\dotsb(\lambda _t+1)_n} \biggl (\frac zt\biggr )^{tn}, \qquad\lambda _1,\dots,\lambda _t\in\mathbb Q\setminus\{-1,-2,\dots\}, $$ where $t$ is an even number, and its derivatives up to the order $t- 1$ inclusive. In the case of algebraic dependence between these functions over $\mathbb C(z)$, a complete structure of algebraic relations between them is given. 
@article{MZM_2002_71_6_a3,
     author = {V. Kh. Salikhov and G. G. Viskina},
     title = {Algebraic {Relations} between the {Hypergeometric} {E-Function} and {Its} {Derivatives}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {832--844},
     publisher = {mathdoc},
     volume = {71},
     number = {6},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a3/}
}
TY  - JOUR
AU  - V. Kh. Salikhov
AU  - G. G. Viskina
TI  - Algebraic Relations between the Hypergeometric E-Function and Its Derivatives
JO  - Matematičeskie zametki
PY  - 2002
SP  - 832
EP  - 844
VL  - 71
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a3/
LA  - ru
ID  - MZM_2002_71_6_a3
ER  - 
%0 Journal Article
%A V. Kh. Salikhov
%A G. G. Viskina
%T Algebraic Relations between the Hypergeometric E-Function and Its Derivatives
%J Matematičeskie zametki
%D 2002
%P 832-844
%V 71
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a3/
%G ru
%F MZM_2002_71_6_a3
V. Kh. Salikhov; G. G. Viskina. Algebraic Relations between the Hypergeometric E-Function and Its Derivatives. Matematičeskie zametki, Tome 71 (2002) no. 6, pp. 832-844. http://geodesic.mathdoc.fr/item/MZM_2002_71_6_a3/

[1] Salikhov V. Kh., “Kriterii algebraicheskoi nezavisimosti znachenii gipergeometricheskikh $E$-funktsii (chetnyi sluchai)”, Matem. zametki, 64:2 (1998), 273–284 | MR | Zbl

[2] Nesterenko Yu. V., Shidlovskii A. B., “O lineinoi nezavisimosti znachenii $E$-funktsii”, Matem. sb., 187:8 (1996), 93–108 | MR | Zbl

[3] Salikhov V. Kh., “Kriterii algebraicheskoi nezavisimosti znachenii odnogo klassa gipergeometricheskikh $E$-funktsii”, Matem. sb., 181:2 (1990), 189–211

[4] Salikhov V. Kh., “Neprivodimost gipergeometricheskikh uravnenii i algebraicheskaya nezavisimosti znachenii $E$-funktsii”, Acta Arith., 53:5 (1990), 453–471 | MR | Zbl

[5] Salikhov V. Kh., “Formalnye resheniya lineinykh differentsialnykh uravnenii i ikh primenenie v teorii transtsendentnykh chisel”, Tr. MMO, 51, URSS, M., 1988, 223–256 | Zbl

[6] Katz N. M., Exponential Sums and Differential Equations, Ann. of Math. Stud., 124, Princeton Univ. Press, Princeton, 1990 | Zbl

[7] Beukers F., Brownawell D., Heckman G., “Siegel normality”, Ann. of Math., 127 (1988), 279–308 | DOI | MR | Zbl

[8] Merzlyakov Yu. I., Ratsionalnye gruppy, Nauka, M., 1987 | Zbl