Fibonacci Number Identities Derived via Nörlund’s Formula

We apply finite hypergeometric sum expressions for Fibonacci numbers related to the work of Dilcher so as to obtain, via Nörlund’s formula for continued fractions, new continued fraction identities in the spirit of Ramanujan. Our applications of Nörlund’s formula allow us to evaluate, in terms of Fibonacci numbers, families of infinite continued fractions that cannot be evaluated by current Computer Algebra Systems. Many of Ramanujan’s continued fractions are such that the partial numerators are given by quadratic polynomials with a free parameter, and the infinite continued fractions that we evaluate in terms of the Fibonacci sequence and that are introduced in our paper are of this form.