[Fwd: two questions on K_\nu]

[Daniel Lozier <dlozier@nist.gov>]

Sender: Federico Pellarin <pellarin@math.unicaen.fr>
Subject: two questions on K_\nu

Here are two questions I cannot solve:

(1) Let $K_{\nu}(z)$ the modified Bessel function of the third
kind (following the usual notations of the Bateman Manuscript Project).
Fix $a$ a real number, let $\nu=ia$ and consider the function
\[f(t)=K_{\nu}(e^{-2\pi t}).\]
I wish to prove, at least for $t$ very big, that $f(t)$
approaches a periodic function of period $1/a$, that I should
study in the sequel.

A similar sub-question arise with the function
\[g(t)=K_{it}(z)e^{\pi t/2},\]
for $z>0$ fixed and $t$ big enough.

Of course, I don't know whether my hypotheses are reasonable,
it's also possible that I'm completely wrong.
Is there anyone knowing the answer?

(2) I found p. 96 of the book "Higher Transcendental Functions"
Vol. 2 a formula (the formula (60) of Nicholson) expressing
the product of two Bessel functions of third kind
$K_{\mu}(Z)K_{\nu}(z)$ as an infinite integral involving
$K_{\mu+\nu}$ "along the real axis".

The question, very naive, is: are there some "similar" formulas
for products of three or more functions such as $K_{\mu}(Z)K_{\nu}(z)
K_{\eta}(\zeta)\ldots $ or at least for $K_{\mu}(z)K_{\nu}(z)
K_{\eta}(z)\ldots $? This formula should not be an "avatar"
of the usual addition theorem. In particular, I'm only considering
integrals of one variable.

Did anyone investigate successfully such a kind of question?

Thank you very much,
Federico Pellarin