Asymptotical solution of a system of linear differential equations with partial derivatives in the case of multiple elementary divisors

1973 ◽  
Vol 24 (2) ◽  
pp. 165-176
Author(s):  
N. I. Shkil' ◽  
V. K. Grigorenko
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
Partial Derivatives ◽  
Elementary Divisors ◽  
Linear Differential

Applications of a theorem by A. B. Shidlovski

1968 ◽  
Vol 305 (1481) ◽  
pp. 149-173 ◽  
Keyword(s):  
Differential Equations ◽  
Bessel Function ◽  
Linear Differential Equations ◽  
Euler’S Constant ◽  
Partial Derivatives ◽  
Arithmetic Properties ◽  
Linear Differential ◽  
Euler's Constant

Shidlovski’s deep theorem on Siegel E -functions satisfying systems of linear differential equations is applied in this paper to the study of the arithmetic properties of the partial derivatives C k (z) = 1/k!{∂/∂v} k J v (z)∣ v=0 ( k = 0,1,2,3) of the Bessel function J 0 (z) . As a by-product, expressions involving Euler’s constant γ and the constant ζ(3) are obtained for which the transcendency can be established.

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