$\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$
Is there a closed form solution for $y=\erf(x+c)+\erf(x-c)$? More specifically, I want to solve
$$\erf(\frac{y}{\sqrt{2}})=\frac{1}{2}\Bigl(\erf\bigl(\frac{x+μ}{\sqrt{2}σ}\bigr) +\erf\bigl(\frac{x-μ}{\sqrt{2}σ}\bigr)\Bigr)$$
Which came out of an optimal transport problem between Normal distributions.
Notes:
- Wolfram Cloud cannot solve it directly.
- Taking derivatives, one gets $y'(x)=e^{-(x+c)^2} + e^{-(x-c)^2}∝\cosh(2cx)e^{-x^2}$
Applying the series expansion $\erf(x)=\frac{2}{\sqrt{π}}\sum_{n=0}^∞ \frac{(-1)^n}{n!(2n+1)}x^{2n+1}$ yields:
$$\begin{aligned} y &=\frac{4}{\sqrt{π}}\sum_{n=0}^∞ \frac{(-1)^n}{n!(2n+1)} \biggl(\sum_{k=0}^{n} \binom{2n+1}{2k}c^{2k}x^{2n+1-2k}\biggr) \\&=\frac{4}{\sqrt{π}}\sum_{n=0}^∞ \frac{(-1)^n}{n!(2n+1)} \biggl(\sum_{j=0}^{n} \binom{2n+1}{2n-2j}c^{2n-2j}x^{2j+1}\biggr) \\&=\frac{4}{\sqrt{π}}\sum_{j=0}^∞\biggl(\sum_{n=j}^∞ \frac{(-1)^n}{n!(2n+1)}\binom{2n+1}{2n-2j}c^{2n-2j}\biggr)x^{2j+1} \\&=\frac{4}{\sqrt{π}}\sum_{j=0}^∞\biggl(\sum_{n=j}^∞ \frac{(-1)^n}{n!(2j+1)}\binom{2n}{2j}c^{2n-2j}\biggr)x^{2j+1} \\&\text{Mathematica says the inner series simplifies to} \\&=\frac{4}{\sqrt{π}}\sum_{j=0}^∞ \frac{(-1)^j {}_1F_1(j+\frac{1}{2}, \frac{1}{2}, -c^2)}{j!(2j+1)}x^{2j+1} \\&\text{where ${}_1F_1$ is a hypergeometric function.} \end{aligned}$$
The last term looks very similar to the series expansion of $\erf(x)$ itself, the $n$-th term gets multiplied by an additional factor $2⋅{}_1F_1(n+\frac{1}{2}, \frac{1}{2}, -c^2)$.
Any suggestions what else I could try? I am looking for a "closed-form" solutions, in terms of other special functions.
EDIT: User Тyma Gaidash٠ points to Marcum-Q function which satisfies
$$Q_{1/2}(x, μ) = \frac{1}{2}\Bigl(\erfc(\frac{x+μ}{\sqrt{2}}) + \erfc(\frac{x-μ}{\sqrt{2}}) \Bigr)$$
Hence, the problem is equivalent to computing an inverse of the Marcum-Q function. This is related to the Noncentral chi-squared distribution; if I understand it correctly, the solution to my problem is given by the quantile function of a non-central χ² distribution.