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 .

**Bell polynomials **

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:

**Entry 1**.

.

**Entry 2**.

.

**Entry 3**.

**Entry 4**.

**Entry 5**.

**Entry 6**.

**Entry 7**.

**Entry 8**.

**Entry 9**.

**Entry 10**.

**Entry 11**.

**Entry 12**.

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