On hyper-sums and hyper-products of progressions

In the article we study properties of some sequences of numbers (so-called “hyper-
sums” and “hyper-products”) which one can construct on the basis of given numerical sequence.

We consider such sequences for arithmetic, geometric progressions and Fibonacci numbers. We
obtain explicit formulas for its calculation and study problems of asymptotic behavior. As a
main result, we prove new asymptotic formula for hyper-products of arithmetic progression that
generalized Stirling’s formula and asymptotic of super-factorial.