Your ONLY path here is to ask the company you travelled with.
They'll be the only one with his details, IF you booked under your names for the trip.
Even then, they likely won't give you his name for privacy reasons.
However, if you impress upon them the gravity of the situation, they might be prepared to contact him on your behalf and see if he's willing to have his contact details shared with you.
travel.stackexchange.com/users/101/Mark Mayo
Contact the coach company:
the man you lent the book to might have handed it in as lost property at the end of his journey.
It's unlikely that they'll be willing to give you his details because that would be a violation of his privacy.
Would they even know who he was?
Do they have your details, for example?
Would you have any way of telling them which of the however-many people on the bus you're trying to contact?
If you can pin down a specific person, you might be able to convince the company to give him your details.
It's also possible that he handed it in as lost property either at the bus station where he got off or to the police.
You could try contacting bus and police stations on the rest of your coach's route.
Does the important document that you've lost identify you?
If so, it's possible that he will find the document, realise its importance and try to find you to give it back.
Otherwise, as suggested in other answers and comments, newspaper ads and social media might help.
But do bear in mind that all of these ideas are long-shots.
travel.stackexchange.com/users/11491/David Richerby
Go to the same coach, on the same day of the week, at the same time.
There's a very good chance he uses it to commute.
You'll probably see him again.
travel.stackexchange.com/users/19276/quemeful
In India Government store the name and other detail only if coach is reserved and not for general people.
If you paid for a reserve seat a call to Rail department will certainly help you.
If it was general coach then you need to take help of Newspaper by publish the ads.
travel.stackexchange.com/users/19283/Anirudha Gupta
Try putting a personal ad in local newspapers that are sold in the towns of origin and destination for your journey.
Maybe you could include TinyURL links (or equivalent) to your Facebook and LinkedIn pages.
It's a long shot, but it may be worth trying if the paper is important enough.
travel.stackexchange.com/users/13160/Erik Kowal
A few days ago, I travelled on an Indian train.  Near my seat there was a guy to whom I have given my book, but forgot to get it back.  There is an important paper inside that book which I need.  How can I find his name and contact number?  Even the name will work as I will search for him on Facebook.  It's entirely my fault - he asked me to wake him up and take it from him and I completely forgot.
How to find out the name of a fellow Indian train passenger?
travel.stackexchange.com
<trains><indian-railways><passenger-rights>
travel.stackexchange.com/questions/35145
travel.stackexchange.com/users/19234/rachita
It is unlikely that the bus company would be willing to give you his details because that would be a violation of his privacy. However, you could contact them or the police stations on the rest of the coach's route in case it was handed in to lost property. Contacting the train company is the recommended method for finding a person you traveleld with. They are unlikely to give you the personàs details but the person may have handed your item into lost property. Other options are contacting the police station or putting an ad in a local paper.
Contact the train company to find a person you travelled with. Alternatively contact police stations. The man who has the book may have handed it in to lost property. The train company is unlikely to give you his details. Put an ad in a local paper.