Линейная независимость значений 𝐸−функций с периодическими коэффициентами

Abstract

We consider sets of integers 𝑎(𝑘,𝑗) 𝑛 , 𝑗 = 1, ..., 𝑚, 𝑘 = 1, ..., 𝑇𝑗 which satisfy conditions

𝑎(𝑘,𝑗)𝑛 = 𝑎(𝑘,𝑗)𝑛+𝑇𝑗, 𝑗 = 1, ..., 𝑚, 𝑘 = 1, ..., 𝑇𝑗 , 𝑛 = 0, 1, ...

and functions

𝐹𝑗,𝑘(𝑧) =∞Σ︁𝑛=0𝑎(𝑘,𝑗)𝑛𝑛!𝑧𝑛, 𝑗 = 1, ..., 𝑚, 𝑘 = 1, ..., 𝑇𝑗 .

We find conditions under which the set of functions

1, 𝑒𝑧, 𝐹𝑗,𝑘(𝑧), 𝑗 = 1, ..., 𝑚, 𝑘 = 2, ..., 𝑇𝑗

is linearly independent over C(𝑧) and for any rational 𝛾 ̸= 0 their values at 𝛾 are linearly
independent numbers.An estimate of the measure of linear independence of these numbers is
obtained. The result can be used to generate pseudo-random numbers.