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$$ \sum_{n=0}^{k+1}\frac{3F_{n+1}-L_{n+1}}{2n!}\frac{(k+1)!}{(k-n+1)!}x^{k-n+1}=(\varphi+x)^k\left(\frac{\sqrt{5}}{5}-\frac{\sqrt{5}-5}{10}x\right)+(\psi+x)^k\left(\frac{\sqrt{5}+5}{10}x-\frac{\sqrt{5}}{5}\right).$$

Wolfram Alpha gave a closed form and after a bit of simplification I got the result above. How would you prove it?

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    $\begingroup$ you may compute the coefficient of $x^{k-n+1}$ in RHS using binomial expansion, and check that it is the same as in LHS using Binet formulae for Fibonacci and Lucas $\endgroup$ – Fedor Petrov Jun 10 at 18:40
  • $\begingroup$ The LHS should be easily writable as a convolution of generating functions for $(3F_i-L_i)/2$ and $k+1\choose i$. $\endgroup$ – Steven Stadnicki Jun 10 at 18:58
  • $\begingroup$ It's just (a+b)^k indeed $\endgroup$ – Pietro Majer Jun 10 at 19:34

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