Experimental identities with Fibonacci series – math.stackexchange.com

Experimental identities with Fibonacci series – math.stackexchange.com

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I obtained the following Fibonacci identities experimentally using SageMath code. Can these be proved theoretically? Identity 1: $$ \sum_{n = 0}^{\infty} \frac{F_{n}x^{n}}{n!} + e^x\sum_{n = 0}^{\...

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