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# Category Archives: Fibonacci series

## Some series on Fibonacci and Lucas numbers

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**.