This paper aims to develop innovative formulae linking Mersenne and Fermat numbers with a class of generalized Fibonacci numbers including the celebrated Fibonacci and Pell numbers. These formulae can be obtained by solving the connection problems between Fermat and generalized Fibonacci polynomials. The introduced connection formulae are obtained with the aid of the power from representations and inversion formulae of Fermat and generalized Fibonacci polynomials. We show that all the connection coefficients are expressed in terms of terminating hypergeometric functions of the type  of certain arguments.

Fibonacci sequence, recurrence relation, hypergeometric functions, connection coefficients.