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 xnk 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 xnk. My hope is that there is an asymptotically linear behaviour of the form
xnk ~ 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...
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)