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} \qquad5. \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.
