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Mat. Sb., 2010, Volume 201, Number 1, Pages 25–58 (Mi msb6395)  

Iterated cyclic exponentials and power functions with extra-periodic first coefficients

A. P. Bulanov

Obninsk State Technical University for Nuclear Power Engineering

Abstract: If $f$ is the iterated $m$-cyclic exponential
$$ f(z)=e^{\lambda\alpha_1ze^{\alpha_2ze^…}}= \langle e^z;\lambda\alpha_1,\alpha_2,…,\alpha_m,\alpha_1,…\rangle, $$
where the first coefficient, $\lambda\alpha_1$, in the sequence of coefficients is extra-periodic, then in its power series expansion at $z=0$, $\sum_{n=0}^\infty\frac1{n!}H^{(n)}(f) z^n$, the form $H^{(n)}(f)$ can be written as
\begin{align*} H^{(n)}(f) &=\lambda\alpha_1\sum_{k_1+…+k_m=n}\frac{n!}{k_1!\dotsb k_m!} (k_1\alpha_2)^{k_2}(k_2\alpha_3)^{k_3}
&\qquad\times…\times(k_{m-1}\alpha_m)^{k_m}[(k_m+\lambda)\alpha_1]^{k_1-1}. \end{align*}
This formula is generalized to any number of extra-periodic coefficients at the start of the sequence. It is also shown that in some cases iterated cyclic exponentials whose first coefficients are not elements of the $m$-cyclic sequence $(\alpha_1,\alpha_2,…,\alpha_m,\alpha_1,…)$ can furnish a solution of a first-order system of differential equations with rational right-hand side.
Bibliography: 32 titles.

Keywords: iterated exponential, cyclic exponential, iterated power function, cyclic power function, coefficient of an exponential, sequence.

DOI: https://doi.org/10.4213/sm6395

Full text: PDF file (709 kB)
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English version:
Sbornik: Mathematics, 2010, 201:1, 23–55

Bibliographic databases:

UDC: 517.521.2+517.537
MSC: 40A30, 30B99
Received: 23.07.2008 and 15.07.2009

Citation: A. P. Bulanov, “Iterated cyclic exponentials and power functions with extra-periodic first coefficients”, Mat. Sb., 201:1 (2010), 25–58; Sb. Math., 201:1 (2010), 23–55

Citation in format AMSBIB
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