I can't find any actual sources, drawings, pictures, or primary sources to verify that Mongolian archers ever actually did that.

If it did happen, I doubt that it ever happened in combat. Here's why:

1. For a bow to be effective, typically it has to be drawn all the way or most of the way. Based on the shape of a horse, rider, and bow, I just don't see how they could draw a bow even half way. I'm tempted to grab my bow (wherever it went), get on my horse, and try it.

2. Every single drawing I've seen (that's just one, but there are others) show horse archers sitting upright while shooting.

3. It just doesn't make sense. Why would you shoot from that position? If your horse is going faster than a walk, leaning that far over and forward greatly increases your chance of falling forward and off the horse. Next, you aren't going to be able to shoot accurately. And finally, you'll probably end up wasting an arrow or hitting your horse in the chin (a horse moves its head a fair amount while galloping).

So, I don't believe it to be something that ever actually happened.

EDIT

So, I wrote up a really long response to somebody who deleted their comment. Here it is:

(He referenced that Lars Anderson archery video).

So, I watched this video twice.

Before anything else, watch this.

When he was shooting from a cramped position, he was only shooting at close range. When he was on the motorcycle, he was shooting exactly how a horse archer would shoot- upright, with the support arm fully extended and the draw hand pulled back to his cheek.

The video does not show the archer doubled over or shooting his bow from what would essentially be the fetal position, which is what you would have to do shoot from the under a horse's chin.

Here is a breakdown of why I don't believe that a mounted archer can shoot from under a horse's chin:

1. The archer would have to ride sideways on the horse. This is not something that is shown in any historical painting, tapestry, statue, or manual that I can find. All of the manuals that I can find tend to say that a rider should stay firmly planted on the horse's back. Now, you cannot steer a horse while sitting on it sideways. I don't care how good of a rider you are. By the basic way horses work, you simply can't steer a horse from that position. Horses move away from pressure. To steer a horse without reins, which you would have to do while shooting from the back of a horse, you need to keep your legs on either side of the animal.

Even if you are OK with your horse running straight ahead, you can't balance the horse while leaning dramatically forward and to one side. If you lean too far to one side, you will throw the horse off balance. If you throw a horse off balance on bad footing (like, mud, grass, non-perfectly level dirt), the horse will trip. This is why modern horseback riders, like three-day eventers, train extensively to learn balance over rough terrain.

If your horse trips while you are leaning far forward, you'll fall forward (because you have stirrups, and the stirrups act as a pivot [Source: Francis Dwyer's manual on cavalry, "Seats and Saddles," from the 1880's. It was true then, and it's true now]), land on your head, and at the very least, get a debilitating concussion.

Not being able to steer and not being able to balance are good enough reasons not to attempt this.

2. I cannot for the life of me figure out how an archer could get a full draw or an accurate shot while shooting tight against a horse's neck, especially while the horse is at a canter or gallop. To get the support hand under the neck of the horse, the rider's legs would have to be over the horse's withers and behind the horse's legs. Let's assume the rider is leaning to the right of the horse. This would mean the rider's left leg is over the horse's withers, and the right leg is under the horse's stomach.

On a side note, this position would greatly inhibit the free range of motion of the horse's right shoulder and front legs, and it would likely make the horse spin.

So, the rider would be holding the bow in his left hand. With his body close enough to the horse's neck to reach the horse's chin with his left hand, he wouldn't be able to bend his left elbow at all. Like, it would require him to bend his elbow backwards to actually reach the chin while still holding on to the bow. Next, in order to hit anything but dirt, he'd have to hold his draw hand out and away from his body, which would greatly limit the amount of power he would get with his draw. Next, he wouldn't be able to see what he was shooting at. And finally, his bow would either be hitting the horse in the chest or in the face. Neither facilitates an accurate or powerful shot.

This already isn't making sense.

3. If the archer is shooting at somebody who can also shoot a bow and arrow, the person NOT bent up like a pretzel is at a huge advantage. The person shooting sensibly can shoot accurately and with power.

If you're bent up and trying to hide behind your horse and shoot me, I'll just shoot your horse in the chest. Your horse will stumble, and because you're as off balance as humanly possible, you'll fall, lose your horse, and probably get a broken neck. I'll then shoot you while you're writhing in pain on the ground or unconscious from a traumatic brain injury. Alternatively, if I am also on a horse and equally skilled, I suppose I could shoot the foot that's still hanging over the horse's withers and watch you fall. Then, I'd shoot you, take your horse, and gain prestige within my community.

4. Mongolian horses are really small. The neck of one wouldn't give any protection. They also aren't tremendously well trained, according to people who have actually ridden them.

In short: There is literally no advantage to a less accurate, less powerful shot from a bad position on a horse. Mongols were good at doing this, and I just don't see them rationally coming to a technique like that. It doesn't make any sense.

Also, here's a video of a Mongolian archer shooting from his horse in a more traditional way. Note how even when he leans a little bit, the horse leans a lot. Notice how he doesn't lean forward at all.

https://www.youtube.com/watch?v=vigGjavgXV4