The general Mittag-Leffler function $$E_{a,b}(z)=\sum_{h=0}^{\infty}\frac{z^h}{\Gamma(ha+b)}$$ satifies the recurrence $$E_{a,b}(z)=zE_{a,b+a}(z)+\frac1{\Gamma(b)}.$$ I am having a hard time in proving this recurrence. Does it follow immediately from definition. I am not getting it.I think the property of Gamma function is used here. Any easy proofs? Thanks beforehand.
2026-03-25 09:33:38.1774431218
Mittag-Leffler function recurrence relation
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Well $\;\displaystyle zE_{a,b+a}(z)=\sum_{h=0}^{\infty}\frac{z^{h+1}}{\Gamma((h+1)a+b)}$ so yes it is immediate.
For other interesting properties of the Mittag-Leffler function see this thread.