Inviting someone to co-author a paper

How I can politely ask him for his contribution?

Just sent him/her an email with a brief summary of the ideas you are interested in and ask him/her if he/she is interested on the project, it has worked for me. Otherwise, try to attend one of his/her talks and approach him/her and the end of the talk, if possible.

How about the contribution, to be more exact, the order of names?

Perhaps it is too early to discuss about this ("Before one brings home the bacon one must kill the pig"). Who knows whether he/she is going to generalise or improve the ideas you propose or he/she would barely be involved in the project (you will come across all sorts of characters while doing research). A safe option is to use the alphabetical order. It helps to avoid many ego-fights. Start with a document without the name of the authors. If you end-up with something worthy of publication, put the cards on the table and ask directly "what order would you be more comfortable with?" or "would you be happy with the order 'ME and John Doe'? ".