No, the degree of the polynomial is ok. We did find two papers though: 1. ASYMPTOTICS OF MODIFIED BESSEL FUNCTIONS OF HIGH ORDER by Sidi and Hoggan (2011), and 2. ASYMPTOTIC EXPANSIONS OF MELLIN TRANSFORMS AND ANALOGUES OF WATSON’S LEMMA by Sidi (1985) which might be sufficient for our purposes.

@Matt: Oh I wish! $\mu$ being purely imaginary is Ramnujan's conjecture.

Nice answer. I wonder if this statement also follows from gauss.math.yale.edu/~ho2/FinalGAFA.pdf (by considering G=SO(n) and H=SO(n-1)).