linear algebra – Recurrence relation with non constant coefficients

I'm studying a true symmetric tridiagonal matrix $$J_ {N + 1}$$ (all non-zero diagonal elements) of dimension $$N + 1$$, and I would like to solve the problem of eigenvalue. The fact is that the coefficients all depend on the dimension $$N + 1$$and changes when N changes.

Diagonal elements are given by $$a (i, N) = – frac {1} {2N} (2i-N) ^ 2$$ or $$(i = 0, …, N)$$, and the off-diagonal elements by $$b (i, N) = – frac {1} {2} sqrt {((N-i-1) i)}$$, or $$i = 1, …, N$$.

Yes $$v$$ is a proper vector, we can write $$J_ {N + 1} v = lambda_ {N + 1} v$$, or $$v = sum_ {i = 0} ^ N mu_ie_i$$ for some coefficients $$mu_i$$ and or $${e_i } _ {i = 0} ^ {N}$$ is a base. The number $$lambda_ {N + 1}$$ (which depends on N) plays the role of eigenvalue of the system.

We find that the coefficients $$mu_i$$ satisfy the following recurrence relation:

$$lambda_ {N + 1} mu_i = b (i, N) mu_ {i + 1} + a (i, N) mu_i + b (i, N) mu_ {i-1}$$,

or $$mu_0 = 1$$ and $$mu _ {- 1} = 0$$.

I am particularly interested in the solution of this relationship in the semi-classical regime where $$N$$ is wide. Maybe things get easier when N tends to infinity.
One possibility to start is to multiply the matrix by $$1 / N$$, so that the coefficients $$a$$ and $$b$$ are linked. We find
$$a (i, N) / N = – frac {1} {2} ( frac {2i} {N} -1) ^ 2$$, and $$b (i, N) / N = -1 / 2 sqrt {(1-i / N-1 / N) (i / N)}$$.

Does anyone have any idea of ​​how to solve this recurrence relationship? Thanks in advance !!