Integration Identities for Some Iterated Exponentials of Order Three

Inspired by the well-known elegant result for the integral of the power tower of length two x x , namely 1 0 1/x x dx = ∞ n=1 1/n n (first attributed to Johann Bernoulli, and commonly referred to as the " sophomore's dream " in present day literature [1]), we present a number of identities for power towers of length three. In the present communication, we shall be primarily concerned with inte-grals of the type I = 1 0 F (x)dx, where the integrand take the form F (x) = T (f (x); g(x); h(x)). Here, T (α; β; γ) = α β γ is a general power tower of length three and f , g and h are sufficiently well behaved functions. Making use of successive exponential transformations, one may write 1 0 (f (x)) a(g(x)) bh(x) dx = ∞ n=0 a n n! ∞ m=0 b n n m m! 1 0 (h(x)) m (ln g(x)) m (ln f (x)) n dx. Applying such a transformation, and making use of the known identity [3] 1 0 x p (ln x) q dx = (−1) q q! (p + 1) q+1 p > −1 , q = 0, 1, 2,. .. , (2) when a = −1, b = 1, f (x) = g(x) = x and h(x) = 1, we recover the identity of Bernoulli as given in the abstract. Similarly, taking a = b = 1, f (x) = g(x) = x and h(x) = 1, one obtains 1 0 x x dx = ∞ n=1 (−1) n+1 /n n. One may obtain even