I hope my answer to be helpful

Best Luck

thank you very much, but i found in " HIGHER TRANSCENDENTAL FUNCTIONS . vol 1 " that this product is a sum of 4F3 hypergeometric functions which his form is not simplified. 

All known results can be found in:

http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/16/02/01/

Thanks all! but it appears that one cannot obtain a single sum to get a simple from more then this known results, then we should use the " Appell Hypergeometric Function "  .

You can find this formula (and similar one) in

Higher Transcendental Functions, Vol. 1

thanks all.

This is a product of two polynomials of degree n and m. Hansen's book, "A Table of Series and Products", Chapter 4, gives a long list of possible ways to re-order, most of which can be worked out by hand if you don't have the book. Depending on the parameters, some of them might simplify. Gradshteyn and Ryzhik, "Table of Integrals Series and Products" , Eq. 0.36 show how to turn this into a single sum with an explicit equation for the coefficients, again as a finite sum. Basically - multiply term by term. I suggest that you use a computer program like Mathematica or Maple.

Thank you for all your suggestions it helps me a lot.

theyt are very importan functions in math generaic functions

Dear Majed Benabdallah

Are you still interested in answering your question? or you are satisfactory?

I have other answer including operational representation

Dear Maged Bin-Saad,

Yes i am interested to have the best answer.

Thank you.

We have got recently a very general expression for the product of two basic hypergeometric series, soon it will be published .