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Trying to solve a PDE coming from the computation of some functional of Brownian motion, I have came across the following Bessel-looking like functional recurrence:

$n^2 g_n(t) + t g_n'(t) = (t/2)(g_{n+1}(t)+g_{n-1}(t))$

where the functions $g_{-n}(t)=g_n(t)$, $n$ integer, $t$ real $\ge0$, are perfectly regular (${}'={\rm d}/{\rm d}t$), and with initial conditions $g_n(0)=\delta_{n,0}$.

Of course, were the term $n^2 g_n(t)$ absent, the solution would simply be $I_n(t)$, the modified Bessel of the second kind. In fact, it is not hard to show that even with this term, the $g_n(t)$'s have the same leading-order asymptotics when $t \to 0$ or $t \to \infty$ as the $I_n(t)$'s.

Anybody has a clue to attack this ? (I have tried Laplace-Mellin transform but was unable to solve the functional recurrence, as well as many others concerning the Taylor coefficients or the Bessel-Neumann ones...)

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3 Answers 3

Have you tried using generating function?

Letting

$$G(x,t)=\sum _{n=0}^{\infty } g_n(t)x^n$$

You can transform the equation into:

$$x G^{(1,0)}(x,t)+x^2G^{(2,0)}(x,t)+t G^{(0,1)}(x,t)=\frac{t}{2}\left(x+\frac{1}{x}\right)G(x,t)$$

You can try to solve this pde and do a series expansion to recover $g_n(t)$

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Thanks but yes! Actually the problem as stated comes from the Fourier coefficients of the solution of the heat-like pde (with initial conditions $G(x,0)=1 \ \forall x$):

$t \partial_t G(x,t) - \partial^2_x G(x,t)-t \cos{x} \ G(x,t)=0$

If the 1st term above were absent, $G$ would be a Mathieu function, and if the 2nd term were missing, $G$ would be the g.f. of the modified Bessel functions $I_n(t)$.

I have just managed to work out the limiting form of the solution when the cosine above is expanded up to second order, because then the ansatz $G_{\rm quad}(x,t) =a(t) e^{-b(t)x^2}$ amounts to solve a Riccati equation with the final result that:

$a(t) = \frac{e^t}{\sqrt{I_0(2 \sqrt{2 t})}}, \ \ \ b(t)=\frac{\sqrt{t}}{2 \sqrt{2}}\frac{I_1(2 \sqrt{2 t})}{I_0(2 \sqrt{2 t})}$

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This differential-difference three-term recursion occurs in conjunction with the study of an imaginary exponential functional of Brownian motion (another question I asked on MO). For those interested with what we were able to find, see http://arxiv.org/abs/1101.1173

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