This question is inspired
by my answer to the question
“How to compute ∏∞n=1(1+1n!)?“.
(for positive integer k)
and I noticed that
f(1)=e−1 was transcendental
(modified Bessel function)
was probably transcendental
(Bessel function) is transcendental.
So, I made the conjecture
that all the f(k)
and I am here presenting it as a question.
The only progress I have made
is to show that
all the f(k) are irrational.
This follows the standard proof that
e is irrational:
and the left side is an integer
and the right side
is an integer plus a proper fraction
I have not been able to prove
but it somehow seems to me
that it should be possible to prove
is not the root of a polynomial
of degree ≤k.
Suppose we have an irreducible polynomial p(X)=adXd+…+a1X+a0∈Z[X] with p(α)=0 and ad≠0.
For each N, we can combine the first N summands and find an integer AN such that
For x close enough to α, we have |p(x)|≤2|ad|⋅|x−α|d, hence for N≫0 and by the MWT,
On the other hand,
is a non-zero integer, hence ≥1 or ≤−1. We conclude
As this inequality cannot hold for infinitely many N, we arrive at a contradiction. We conclude that p as above does not exist and so α is transcendental.