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I have recently come across an ODE of the form

$$y''+(a+b x^2)y'+(c+dx+h/x^2)y=0 \hspace{30mm} (*)$$

where $y=y(x)$ and $a,b,c,d,h$ are arbitrary constants.

As far as I understand (please correct me if I'm wrong) for $h=0$ we get a special case of the triconfluent Heun equation but I wonder what is known for nonzero $h$.

In particular, I would like to know the following (for nonzero $h$) :

1) can one express the solution of (*) in terms of some known special functions, and, if yes, of which ones?

2) As far as I see for special values of parameters there should exist polynomial solutions of any natural degree (one just assumes $y(x)$ to be Taylor series and then truncates at degree $n$). I am almost certain that these polynomials were already studied but was not able to find them in the literature. So, do they belong to some known class, and, if yes, which class is that?

Special thanks for the relevant references.

P.S. More general case of (*) with $h/x^2$ replaced by $h x^\alpha$ with integer $\alpha\neq 0,1$ would be of interest too.

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  • $\begingroup$ Maple has tried hard to include the Heun functions. It does, indeed, solve your equation with $h=0$ in terms of the Heun triconfluent function. But Maple does not know how to solve your equation with the $h$. But it does solve when you have $hx^2$ or $hx^3$ or $hx^4$. But not $hx^5$. $\endgroup$ – Gerald Edgar Oct 26 '16 at 13:49
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A small case $hx^2$, as mentioned in the comment. According to Maple,

picture

the solution of
diff(y(x),x,x)+(1+b*x^2)*diff(y(x),x)+(c+d*x+h*x^(2))*y(x);
is
y(x) = _C1*HeunT(3^(2/3)*(b^2*c-b*h+h^2)/b^(8/3), -3*(b-d)/b, (b-2*h)*3^(1/3)/b^(4/3), (1/3)*3^(2/3)*b^(1/3)*x)*exp(-(1/3)*(b^2*x^2+3*b-3*h)*x/b)+_C2*HeunT(3^(2/3)*(b^2*c-b*h+h^2)/b^(8/3), 3*(b-d)/b, (b-2*h)*3^(1/3)/b^(4/3), -(1/3)*3^(2/3)*b^(1/3)*x)*exp(-x*h/b)

where $HT$ or HeunT is the Heun triconfluent function

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  • $\begingroup$ Thanks a lot! Sorry I don't have upvote privileges yet. It's a pity nothing seems to be known about the $h/x^2$ case. $\endgroup$ – just-someone Oct 26 '16 at 15:35

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