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In this post, we shall see several curious summation formulas of the Fibonacci numbers and the related Lucas numbers that involves the Bell polynomials and the incomplete gamma function . The Fibonacci numbers are defined as and the Lucas numbers are defined as .
The Bell polynomials of polynomials are defined by the exponential generating function
The first few Bell polynomials are
Incomplete gamma function
The incomplete gamma function is defined as
Entry 1-6 involve summation identities involving the Bell polynomials or the incomplete gamma function.
As usual, denotes golden ratio. We have the following results: