On Zudilin's q-question about Schmidt's problem

Details On Zudilin's q-question about Schmidt's problem On Zudilin's q-question about Schmidt's problem

2012-04-01

arxiv.org/abs/1204.0187v2

Mathematics

Combinatorics

5 pages, two open problems are added

Scientific paper

For any integer $r\geqslant 2$, using the $q$-Pfaff-Saalsch\"utz identity, we prove that there exists a (unique) sequence of Laurent polynomials $\{b^{(r)}_k(q)\}_{k=0}^\infty$ in $q$ with nonnegative integral coefficients such that \sum_{k=0}^n q^{-rnk} {n\brack k}^r {n+k\brack k}^r =\sum_{k=0}^n q^{-nk} {n\brack k}{n+k\brack k}b^{(r)}_k(q), where ${n\brack k}$ denotes the $q$-binomial coefficient. This gives a new solution to Zudilin's question about finding a $q$-analogue of Schmidt's problem.

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