I know little about spheroidal functions and browsing Wolfram functions, the Spheroidal Eigenvalue function $\lambda_{n,m}(z)$ was intriguing. According to the DLMF section 30.3(iii):
$$b_p-\lambda-\frac{a_{p-2}c_p}{b_{p-2}-\lambda-} \frac{a_{p-4}c_{p-2}}{b_{p-4}-\lambda-}\cdots= \frac{a_pc_{p+2}}{b_{p+2}-\lambda-} \frac{a_{p+2}c_{p+4}}{b_{p+4}-\lambda-}\cdots\\a_k=-(k+1)(k+2),b_k=(m+k)(m+k+1)-c^2,c_k=c^2$$ If $p$ is even, then $\lambda=\lambda_{m+2\Bbb N,m}(c^2)$ and if $p$ is odd, then $\lambda=\lambda_{m+2\Bbb N+1,m}(c^2)$
Using continued fraction K notation:
$$b_p-\lambda-\frac{a_{p-2}c_p}{b_{p-2}-\lambda-}\cdots \frac{a_{p-2t}c_{p-2(t-1)}}{b_{p-2t}-\lambda-}\cdots= \frac{a_pc_{p+2}}{b_{p+2}-\lambda-} \cdots\frac{a_{p+2(t-1)}c_{p+2t}}{b_{p+2t}-\lambda-} \cdots\\ b_p-\lambda-\mathop{\text K}_{t=1}^\infty \frac{-a_{p-2t}c_{p-2(t-1)}}{b_{p-2t}-\lambda}= \mathop{\text K}_{t=1}^\infty \frac{-a_{p+2(t-1)}c_{p+2t}}{b_{p+2t}-\lambda}\\(m+p)(m+p+1)-c^2-\lambda-\mathop{\text K}_{t=1}^\infty \frac{(p-2t+1)(p-2t+2)c^2}{(m+p-2t)(m+p-2t+1)-c^2-\lambda}= \mathop{\text K}_{t=1}^\infty \frac{(p+2t-1)(p+2t)c^2}{(m+p+2t)(m+p+2t+1)-c^2-\lambda} $$
Hopefully this expansion is correct. If it is, then there is no closed form. Similarly, papers like this one with formula $(A.3)$ define a new function for the continued fraction.
However, a special case of the spheroidal eigenvalue function are the Mathieu characteristics which gives the inverse of the Mathieu characteristic exponent, which is not the argument of a Mathieu function. The continued fractions may be expressible in terms of a general special function.
It is definable using the angular spheroidal function of the first kind $\text{PS}_{n,m}(c,z)$, its derivative wrt $z$ from this formula, and similar ones from other spheroidal functions:
$$\frac{2z\text{PS}’_{n,m}(c,z)-(1-z^2)\text{PS}’’_{n,m}(c,z)}{\text{PS}_{n,m}(c,z)}-(1-z^2)c^2+\frac{m^2}{1-z^2} =\lambda_{n,m}(c)$$
but then we need to know what $\frac{d^2\text{PS}_{n,m}(c,z)}{dz^2}$ is.
Special Cases:
$$x^3-8x^2+4(4c^2+3)x-64c^2=0\implies x=\lambda_{n,3}(c),n=-3,-2,-1,0,1,2,3$$ partly shown here
We can find other equations $\lambda_{n,m}(z)$ solves for general $n,m,z$, but to narrow down:
What is a special case, closed form, or integral/sum form of the continued fraction?