**49.** We have lim_{z → q}(*z*^{km + r} – 1)/(*z*^{lm + r} – 1) = 1 when 0 < *r* < *m*, and the limit is lim_{z → q}(*kmz*^{km – 1})/(*lmz*^{lm – 1}) = *k*/*l* when *r* = 0. So we can pair up factors of the numerator Π_{n – k < a ≤ n}(*z*^{a} – 1) with factors of the denominator Π_{0 < b ≤ k}(*z ^{b}* – 1) when

*Notes:* This formula was discovered by G. Olive, *AMM* **72** (1965), 619. In the special case *m* = 2, *q* = –1, the second factor vanishes only when *n* is even and *k* is odd.

The formula holds for all *n* ≥ 0, but is *not* always equal to . The reason is that the second factor is zero unless ...

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