I am interested in the asymptotics for large order and large argument for the modified Bessel function of first kind $$I_{\frac{N-1}{2}} (Nz) $$ where
- $z$ is a fixed positive real number $z>0$ and
- $N$ is natural number.
I want to pass $N$ to $+\infty$. I checked the NIST collection and the book by Abramowitz and Stegun, but I did not find anything useful. Do you have any tips? Thank you very much.
Let $\kappa \in \mathbb C$ be fixed, $x>0$ and $\operatorname{Re}(\nu)>0$. By $(10.32.12)$, we can write $$ I_{\nu + \kappa } (\nu \operatorname{csch} x) = \frac{1}{{2\pi {\rm i}}}\int_{\infty - {\rm i}\pi }^{\infty + {\rm i}\pi } {\exp \left( { - \nu (t - \operatorname{csch} x\cosh t)} \right)\,{\rm e}^{ - \kappa t} {\rm d}t} . $$ The relevant saddle point is at $t=x$. An application of the saddle point method then gives $$ I_{\nu + \kappa } (\nu \operatorname{csch}x) \sim \frac{{\exp (\nu (\coth x - x) - \kappa x)}}{{\sqrt {2\pi \nu \coth x} }}\sum\limits_{n = 0}^\infty {\frac{{U^{\kappa}_n (\tanh x)}}{{\nu ^n }}} , $$ as $\nu \to \infty$ in the sector $|\arg \nu|\le \frac{\pi}{2}-\delta<\frac{\pi}{2}$, uniformly with respect to $x>0$ (cf. $(10.41.3)$). The coefficients $U^{\kappa}_n(w)$ are polynomials in $w$ of degree $3n$, the first few being \begin{align*} & U^{\kappa}_0(w)=1,\\ & U^{\kappa}_1(w)= - \frac{5}{{24}}w^3 - \frac{\kappa }{2}w^2 + \frac{{1 - 4\kappa ^2 }}{8}w, \\& U^{\kappa}_2(w)=\frac{{385}}{{1152}}w^6 + \frac{{35\kappa }}{{48}}w^5 + \frac{{7(20\kappa ^2 - 11)}}{{192}}w^4 + \frac{{\kappa (20\kappa ^2 - 29)}}{{48}}w^3 + \frac{{16\kappa ^4 - 40\kappa ^2 + 9}}{{128}}w^2 . \end{align*} In general, $$ U^{\kappa}_n(w) = \frac{{( - w)^n }}{{2^n n!}}\left[ {\frac{{{\rm d}^{2n} }}{{{\rm d}t^{2n} }}\left( {{\rm e}^{ - \kappa t} \left( {\frac{1}{2}\frac{{t^2 }}{{\cosh t - 1 + w(\sinh t - t)}}} \right)^{n + 1/2} } \right)} \right]_{t = 0} . $$ In this paper, the following recurrence was established: $$ U_{n + 1}^\kappa (w) = - \kappa w^2 U_n^\kappa (w) + \frac{1}{2}w^2 (1 - w^2 )\frac{{{\rm d}U_n^\kappa (w)}}{{{\rm d}w}} + \frac{1}{8}\int_0^w {(1 - 4\kappa ^2 + 8\kappa t - 5t^2 )U_n^\kappa (t){\rm d}t} $$ for $n\ge 0$, with initial value $U^{\kappa}_0(w)=1$.
In your case $\nu = \frac{N}{2}$, $x = \operatorname{csch}^{ - 1} (2z)$, and $\kappa = - \frac{1}{2}$.