<Dd> (1 − z) − u = ∑ k = 0 ∞ u k ∑ n = k ∞ z n n! (n k) = e u log ⁡ (1 / (1 − z)) (\ displaystyle (1 - z) ^ (- u) = \ sum _ (k = 0) ^ (\ infty) u ^ (k) \ sum _ (n = k) ^ (\ infty) (\ frac (z ^ (n)) (n!)) \ left ((n \ atop k) \ right) = e ^ (u \ log (1 / (1 - z)))) </Dd> <Dl> <Dd> log m ⁡ (1 + z) 1 + z = m! ∑ k = 0 ∞ s (k + 1, m + 1) z k k!, m = 1, 2, 3,...z <1 (\ displaystyle (\ frac (\ log ^ (m) (1 + z)) (1 + z)) = m! \ sum _ (k = 0) ^ (\ infty) (\ frac (s (k + 1, m + 1) \, z ^ (k)) (k!)), \ qquad m = 1, 2, 3, \ ldots \ quad z <1) </Dd> </Dl> <Dd> log m ⁡ (1 + z) 1 + z = m! ∑ k = 0 ∞ s (k + 1, m + 1) z k k!, m = 1, 2, 3,...z <1 (\ displaystyle (\ frac (\ log ^ (m) (1 + z)) (1 + z)) = m! \ sum _ (k = 0) ^ (\ infty) (\ frac (s (k + 1, m + 1) \, z ^ (k)) (k!)), \ qquad m = 1, 2, 3, \ ldots \ quad z <1) </Dd> <Dl> <Dd> ∑ n = i ∞ (n i) n (n!) = ζ (i + 1), i = 1, 2, 3,...(\ displaystyle \ sum _ (n = i) ^ (\ infty) (\ frac (\ left ((n \ atop i) \ right)) (n \, (n!))) = \ zeta (i + 1), \ qquad i = 1, 2, 3, \ ldots) </Dd> </Dl>

Stirling numbers of the first kind recurrence relation