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It is well-known that ( − log(1 − z)) l = l! ∑ z nl n1n2 · · · nl nl>···>n1≥1 for |z | < 1 and l any positive integer. The author proves two generalisations of this identity: zn) l = l! k + α znl (n1 + α)(n2 + 2α) · · · (nl + lα) n=1 nl>...n1≥1 (for |z | < 1, α ≥ 0 and l any positive integer) and an explicit power series expansion of powers of the q–analogue of log defined by ∞ ∑ znqn 1 − qn n=1 (for |z | ≤ 1 and |q | < 1). The second identity is too complicated to state but also involves certain multiple finite sums. When α = 0 or when q → 1 (in this case, after multiplication by (1 − q) l), one recovers the identity for ( − log(1 − z)) l. The proof of the first generalisation is by induction on l while the proof of the second is based on the following interesting combinatorial identity: 1 (x1 − 1)(x2 − 1) · · · (xl − 1) wher

Year: 2013

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