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Looking for self-similar, radial solutions to a certain nonlinear wave equation in $\mathbb{R}^{d+1}$ leads to studying the following linear ODE in the self-similar variable $a\in[0,1)$: $$(a^2-1)Q''(a) + \left(3a-\frac{d-1}{a} \right)Q'(a) + \frac{3}{4}Q(a)=0$$ Mathematica tells me that the solution set is spanned by $$\operatorname{span}\left\{ {}_2F_1(1/4, 3/4, d/2, a^2),a^{2-d}i^{2-d}{}_2F_1(5/4-d/2, 7/4-d/2, 2-d/2, a^2)\right\},$$ where ${}_2F_1$ is the Gauss hypergeometric function. I have two questions:

  1. The second hypergeometric function is undefined for even $d\ge 4$ as then $2-d/2$ is a nonpositive integer. What happens then to the set of solutions?

  2. For odd $d\ge 3$, by checking each case individually on Mathematica, I want to conjecture that the hypergeometric solutions simplify so that the set of solutions is given by $$\sum_{\pm}c_{\pm}a^{2-d}(1\pm a)^{d/2-1}P^d_{\pm}(a),$$ where $P^d_\pm(a)$ is a polynomial that is non-vanishing at $a\in\{-1, 1\}$ of degree $d/2-3/2\in \mathbb{N}_0$. Is this conjecture true or false?

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  • $\begingroup$ Seems that the last term should read $\frac 3 4 Q(a)$ instead of $\frac 3 4 Q'(a)$ (also in the title). $\endgroup$
    – Fred Hucht
    Commented 2 hours ago
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    $\begingroup$ Next, you can substitute $a\mapsto \sqrt{b}$, $Q(a)\mapsto f(b)$ to get $4 (b-1) b f''(b)+(8 b-2 d) f'(b)+\frac{3}{4}f(b) = 0$, with solutions in terms of Legendre $P$ and $Q$. These solutions are also defined for $d\geq4$. $\endgroup$
    – Fred Hucht
    Commented 2 hours ago
  • $\begingroup$ @FredHucht. Thank you, it it now fixed. $\endgroup$
    – Dispersion
    Commented 1 hour ago

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