n X k=0 1 1 − qk+m · q−rkQn j=1(1 − qj) Qk−1 j=0(1 − qj−k)Qn j=k+1(1 − qj−k

Problem 12183

(American Mathematical Monthly, Vol.127, May 2020) Proposed by H. Ohtsuka (Japan).

For integers m, n, and r with m ≥1 and n ≥ r ≥ 0, prove

n

X

k=0

(−1)^kq(^k+12 )−rk

1 − q^k+m

n k



q

= q^rm 1 − q^m

m + n m

−1 q

wheren k



q denotes the Gaussian binomial coefficient.

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

Solution. We consider the partial fraction decomposition of the left-hand side:

q^rm 1 − q^m

m + n m

−1 q

=q^rmQn

j=1(1 − q^j) Qn

j=0(1 − q^j+m)

=

n

X

k=0

1

1 − q^k+m · q^−rkQn

j=1(1 − q^j) Qk−1

j=0(1 − q^j−k)Qn

j=k+1(1 − q^j−k)

=

n

X

k=0

1 1 − q^k+m ·

q^−rkQn

j=1(1 − q^j) Qk

j=1(1 − q^−j)Q_n−k

j=1(1 − q^j)

=

n

X

k=0

1

1 − q^k+m · q^−rkQn

j=1(1 − q^j) (−1)^kq⁻(^k+12 ) Q^k

j=1(1 − q^j)Qn−k

j=1(1 − q^j)

=

n

X

k=0

(−1)^kq(^k+12 )−rk

1 − q^k+m

n k



q

.