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

作者：Bulanov A P^*

来源：Sbornik: Mathematics , 2010, 201(1-2): 23-55.

DOI：10.1070/SM2010v201n01ABEH004064

• iterated exponential
• cyclic exponential
• iterated power function
• cyclic power function
• coefficient of an exponential
• sequence

If f is the iterated m-cyclic exponential
f (z) = e(lambda alpha 1ze alpha 2ze...) = < e(z); lambda alpha(1), alpha(2),...,alpha(m), alpha(1),...>,
where the first coefficient, lambda(alpha 1), in the sequence of coefficients is extra-periodic, then in its power series expansion at z = 0, Sigma(infinity)(n=0) (1/n!)H((n))(f)z(n), the form H((n))(f) can be written as
H((n))(f) = lambda alpha(1) Sigma(k1+...+km=n) n!/k(1)...k(m)!(k(1)alpha(2))k(2)(k(2)alpha(3))k(3)
X ... X (k(m)-1 alpha(m))k(m)[(k(m) + lambda)alpha(1)](k1-1).
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.

• 出版日期2010-2