$1.$ $\displaystyle e^{-x^2/2}$ (more generally $e^{-x^2/2}H_{2n}(x)$, $H_n$ is Hermite polynomial)
$2.$ $\displaystyle \frac{1}{\sqrt{x}}$ $\qquad$ $3.$ $\displaystyle\frac{1}{\cosh\sqrt{\frac{\pi}{2}}x}$ $\qquad$ $4.$ $\displaystyle \frac{\cosh \frac{\sqrt{\pi}x}{2}}{\cosh \sqrt{\pi}x}$ $\qquad$$5.$ $\displaystyle\frac{1}{1+2\cosh \left(\sqrt{\frac{2\pi}{3}}x\right)}$
$6.$ $\displaystyle \frac{\cosh\frac{\sqrt{3\pi}x}{2}}{2\cosh \left( 2\sqrt{\frac{\pi}{3}} x\right)-1}$ $\qquad$ $7.$ $\displaystyle \frac{\cosh\left(\sqrt{\frac{3\pi}{2}}x\right)}{\cosh (\sqrt{2\pi}x)-\cos(\sqrt{3}\pi)}$ $\qquad$ $8.$ $\displaystyle \cos\left(\frac{x^2}{2}-\frac{\pi}{8}\right) $
$9.$ $\displaystyle\frac{\cos \frac{x^2}{2}+\sin \frac{x^2}{2}}{\cosh\sqrt{\frac{\pi}{2}}x}$ $\qquad$ $10.$ $\displaystyle \sqrt{x}e^{-\frac{1}{2}x^2\cos\theta}J_{-\frac{1}{4}}\left(\frac{x^2}{2}\sin\theta\right)$ $\qquad$ $11.$ $\displaystyle \frac{\sqrt[4]{a}\ K_{\frac{1}{4}}\left(a\sqrt{x^2+a^2}\right)}{(x^2+a^2)^{\frac{1}{8}}}$
$12.$ $\displaystyle \frac{x e^{-\beta\sqrt{x^2+\beta^2}}}{\sqrt{x^2+\beta^2}\sqrt{\sqrt{x^2+\beta^2}-\beta}}$$\qquad$ $13.$ $\displaystyle \psi\left(1+\frac{x}{\sqrt{2\pi}}\right)-\ln\frac{x}{\sqrt{2\pi}}$, $\ \psi$ is digamma function.
$14.$ $f(x)=\begin{cases} \displaystyle 0,\qquad\qquad \qquad\qquad\quad 0 < x < z\\ \displaystyle \frac{xJ_{-\frac{3}{4}}\large(z\sqrt{x^2-z^2}\large)}{\large(x^2-z^2\large)^{\frac{3}{8}}},\qquad\ \ \ z < x \end{cases}$ $\qquad$ $15.$ $\displaystyle e^{\frac{x^2}{4}}K_{0}\left(\frac{x^2}{4}\right)$
Examples $1-5,8-10$ are from the chapter about self-reciprocal functions in Titschmarsh's book "Introduction to the theory of Fourier transform". Examples $11$, $12$, $14$ and $15$ can be found in Gradsteyn and Ryzhik. Examples $6$ and $7$ are from this question What are all functions of the form $\frac{\cosh(\alpha x)}{\cosh x+c}$ self-reciprocal under Fourier transform?. Some other self-reciprocal functions composed of hyperbolic functions are given in Bryden Cais's paper On the transformation of infinite series. Discussion of $13$ can be found in Berndt's article.
$\bf{2.}$ Eigenfunctions of the sine Fourier transform $\sqrt{\frac{2}{\pi}}\int_0^\infty f(x)\sin ax dx=f(a)$:
$1.$ $\displaystyle \frac{1}{\sqrt{x}}$ $\qquad$ $2.$ $\displaystyle xe^{-x^2/2}$ (and more generally $e^{-x^2/2}H_{2n+1}(x)$)
$3.$ $\displaystyle \frac{1}{e^{\sqrt{2\pi}x}-1}-\frac{1}{\sqrt{2\pi}x}$ $\qquad$ $4.$ $\displaystyle \frac{\sinh \frac{\sqrt{\pi}x}{2}}{\cosh \sqrt{\pi}x}$ $\qquad$ $5.$ $\displaystyle \frac{\sinh\sqrt{\frac{\pi}{6}}x}{2\cosh \left(\sqrt{\frac{2\pi}{3}}x\right)-1}$
$6.$ $\displaystyle \frac{\sinh(\sqrt{\pi}x)}{\cosh \sqrt{2\pi} x-\cos(\sqrt{2}\pi)}$ $\qquad$ $7.$ $\displaystyle \frac{\sin \frac{x^2}{2}}{\sinh\sqrt{\frac{\pi}{2}}x}$ $\qquad$ $8.$ $\displaystyle \frac{xK_{\frac{3}{4}}\left(a\sqrt{x^2+a^2}\right)}{(x^2+a^2)^{\frac{3}{8}}}$
$9.$ $\displaystyle \frac{x e^{-\beta\sqrt{x^2+\beta^2}}}{\sqrt{x^2+\beta^2}\sqrt{\sqrt{x^2+\beta^2}+\beta}}$$\qquad$ $10.$ $\displaystyle \sqrt{x}e^{-\frac{1}{2}x^2\cos\theta}J_{\frac{1}{4}}\left(\frac{x^2}{2}\sin\theta\right)$
$11.$ $\displaystyle e^{-\frac{x^2}{4}}I_{0}\left(\frac{x^2}{4}\right)$ $\qquad$ $12.$ $\displaystyle \sin\left(\frac{3\pi}{8}+\frac{x^2}{4}\right)J_{0}\left(\frac{x^2}{4}\right) $$\qquad$ $13.$ $\displaystyle \frac{\sinh \sqrt{\frac{2\pi}{3}}x}{\cosh \sqrt{\frac{3\pi}{2}}x}$
$14.$ $ f(x)=\begin{cases} \displaystyle 0,\qquad\qquad \qquad\qquad\quad 0 < x < z\\ \displaystyle \frac{J_{-\frac{1}{4}}\large(z\sqrt{x^2-z^2}\large)}{\large(x^2-z^2\large)^{\frac{1}{8}}},\qquad\ \ \ z < x \end{cases} $
$15.$ $\displaystyle{\frac{\tanh\sqrt{\frac{\pi}{8}}x}{\cosh\sqrt{\frac{\pi}{2}}x}} $ $\qquad$ $16.$ $\displaystyle{\frac{\sinh\sqrt{\frac{\pi}{10}}x\sinh\sqrt{\frac{2\pi}{5}}x}{\sinh\sqrt{\frac{5\pi}{2}}x}} $
$17.$ $ \int\limits_0^\infty\cfrac{\sin \pi nx/2}{x+\cfrac{1^2}{x+}\cfrac{2^2}{x+}\cfrac{3^2}{x+\text{...}}}dx=\cfrac{1}{n+\cfrac{1^2}{n+}\cfrac{2^2}{n+}\cfrac{3^2}{n+\text{...}}} $
$18.$ $ \int\limits_0^\infty\cfrac{\sin nx}{x+\cfrac{1}{x+}\cfrac{2}{x+}\cfrac{3}{x+\text{...}}}dx=\cfrac{\sqrt{\pi/2}}{n+\cfrac{1}{n+}\cfrac{2}{n+}\cfrac{3}{n+\text{...}}} $
Examples $1-5,7$ can be found in Titschmarsh's book cited above. $8-12$ and $14$ can be found in Gradsteyn and Ryzhik. $13$ is from Bryden Cais, On the transformation of infinite series, where more functions of this kind are given. Continued fractions are due to Ramanujan.
$\bf{3.}$ Eigenfunctions of the Hankel transform of order $\nu$, $\int_0^\infty f(x)xJ_{\nu} (ax) dx=f(a)$:
$1.$ $\displaystyle J_{\frac{\nu}{2}} \left(\frac{x^2}{2}\right)$ $\qquad$ $2.$ $ \displaystyle x^{\nu}\frac{K_{\frac{\nu+1}{2}}\large(a\sqrt{x^2+a^2}\large)}{\large(x^2+a^2\large)^{\frac{\nu+1}{4}}}$
$3.$ $f(x)=\begin{cases} \displaystyle 0,\qquad\qquad \qquad\qquad\quad 0 < x < z\\ \displaystyle x^{-\nu}\large(x^2-z^2\large)^{\frac{\nu-1}{4}}J_{\frac{\nu-1}{2}}\large(z\sqrt{x^2-z^2}\large),\qquad\ \ \ 0 < z < x \end{cases}$
Examples $1-3$ are from Titschmarsh's book cited above.
