When sound waves diffract through a single slit, do they produce an interference pattern which is mathematically identical to that of light waves ...?

The answer is no. The diffraction pattern of sound and of light behind a slit is not similar and so the mathematical description is not complete. This is because the real patterns on an observation screen (or an imaginery line for sound waves) are never the same for sound and light. The fringes from the light diffraction are not moving patterns. More, the reflection of these light (from the observation screen) does not interact with the incomming waves. The diffracted water (or sound) waves don't stand still (picture), they are moving to the left and to the right if one look on them at the position there is the screen for light fringes. If one put a screen in the reflected waves interact with the waves from the slit and produce more complicated patterns.

We have to state that the mathematical model describes for water waves a instantaneous situation. The position of the spherical water waves behind a slit depends from the postion of the wave in front of the slit at some time. Opposite situation for light, the fringes from light stand still.

As pointed out in the answer from Ben Crowell, the slit can work as a waveguide. As pointed out from Floris, the edges could "electrically conducting at the frequency of interest for the electromagnetic case". In both cases the light interacts with the slits material. This do the light with every edge too, fringes appear behind every edge. As I pointed out here, "light is a lot of photons, and if you take monochromatic light, a lot of same energy photons and they interact with the electric field of the electrons of the edge. The last point one has to imagine is that the field they "built" together is quatitized. The fringes on the observers screen show how the field is quantized."

So a mathematical model for sound or water can predict instantaneous distances between maximum and minimum points of the sperical wave behind a slit. And a matematical model for light can predict the positions of maxima and minima for fringes on an observation screen. But physically that are total different phenomena.

When sound waves diffract through a single slit, do they produce an interference pattern which is mathematically identical to that of light waves ...?

The answer is no. The diffraction pattern of sound and of light behind a slit is not similar and so the mathematical description is not complete. This is because the real patterns on an observation screen (or an imaginery line for sound waves) are never the same for sound and light. The fringes from the light diffraction are not moving patterns. More, the reflection of these light (from the observation screen) does not interact with the incomming waves. The diffracted water (or sound) waves don't stand still (picture), they are moving to the left and to the right if one look on them at the position there is the screen for light fringes. If one put a screen in the reflected waves interact with the waves from the slit and produce more complicated patterns.

We have to state that the mathematical model describes for water waves a instantaneous situation. The position of the spherical water waves behind a slit depends from the postion of the wave in front of the slit at some time. Opposite situation for light, the fringes from light stand still.

As pointed out in the answer from Ben Crowell, the slit can work as a waveguide. As pointed out from Floris, the edges could "electrically conducting at the frequency of interest for the electromagnetic case". In both cases the light interacts with the slits material. This do the light with every edge too, fringes appear behind every edge. As I pointed out here, "light is a lot of photons, and if you take monochromatic light, a lot of same energy photons and they interact with the electric field of the electrons of the edge. The last point one has to imagine is that the field they "built" together is quatitized. The fringes on the observers screen show how the field is quantized."

So a mathematical model for sound or water can predict instantaneous distances between maximum and minimum points of the sperical wave behind a slit. And a matematical model for light can predict the positions of maxima and minima for fringes on an observation screen. But physically that are total different phenomena.