For every (fixed) given integer k = 0,1,2,..., consider the following real valued function of x:

     f(x,k) = exp(x)*HeunC(2*x, 1/2, -1/2, -x^2, k*(k+1) + 1/8, 99/100)

Let x_nk denote the n-th positive zero of f(x,k), for n=1,2,3,... Then I am interested to find the asymptotic behaviour of x_nk. My hope is that there is an asymptotically linear behaviour of the form

     x_nk ~ a + b*n + c*k

with real constants a,b,c.

Note: The last argument "99/100" in the function above is thought to be an approximation for the limit to 1 from below. For instance, I would prefer a value 0.9999999...  

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EDIT: The plot below is what I could reach with maple without to get instabilities (l=k)

This result was obtained with:

(Sorry, but I don't know how to insert the true Maple code)

I would like to compute the function 

                       f(x) := exp(x) * HeunC(2*x, 1/2, -1/2, -x^2, 1/8, 99/100)

for very large real values of x. However, in the following plot it can be seen that there are already strong numerical instabilities for x>33. What can I do to reach more stability for large x? Or, does somebody know how to get a closed form asymptotic expression for large x?