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For $varepsilon>0$ consider the tridiagonal matrix

$$L_{varepsilon}=begin{bmatrix}
0 & 1 & & & & & & \
1 & varepsilon & 1 & & & & & \
& 1 & 2 varepsilon & 1 & & & \
& & 1 & 3 varepsilon & 1 & & & \
& & & 1 & 4 varepsilon & 1 & & \
& & & & ddots & ddots & ddots & \
end{bmatrix}$$

with all blank entries equal to zero. A column vector $$Phi = begin{bmatrix} Phi_0 \ Phi_1 \ Phi_2 \ vdots end{bmatrix}$$ is an eigenvector $$L_{varepsilon} Phi = u Phi$$ iff the entries $Phi_h = Phi_{h}(u; varepsilon)$ satisfy a three-term recurrence $$Phi_{h-1} (u; varepsilon) + h varepsilon Phi_h(u; varepsilon) + Phi_{h+1}(u; varepsilon) = uPhi_{h}(u;varepsilon)$$ for $h geq 0$ with convention $Phi_{-1} = 0$, which is equivalent to
$$Phi_{h-1} (u; varepsilon) + (h varepsilon - u) Phi_h(u; varepsilon) + Phi_{h+1}(u; varepsilon) = 0.$$

Question: Can one identify the $Phi_h(u; varepsilon)$ with a known function, possibly a specialization or degeneration of the Gauss hypergeometric function?

As $varepsilon rightarrow 0$, the diagonal in $L_{varepsilon}$ disappears and $Phi_{h}(u; 0)$ are the Chebyshev polynomials. The recurrence above seems close to that of the Gegenbauer polynomials but is not exactly the same. Any help would be greatly appreciated!