I have tried to prove it expanding the power series but it seems very complex and not doable. Is there any simple trick to prove this?
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1It is doable using power series: remember the binomial expansion for $(z_1+z_2)^n$. – Angina Seng Sep 02 '18 at 10:46
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yes, but it is very difficult. – pie Sep 02 '18 at 10:46
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1Really, it isn't. – Angina Seng Sep 02 '18 at 10:47
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Maybe consider the Cauchy summation? – Vim Sep 02 '18 at 10:47
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@Vim No, Cauchy summation has nothing to do with it. It's entirely straighforward: Write $e^{z_1+z_2}$ as a power series and then apply the binomial theorem to $(z_1+z_2)^n$. – David C. Ullrich Sep 02 '18 at 14:59