By Krattenthaler C.

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Li + A − n]q ! [A − 2i − 1]q ! 16) [Li − B + n]q ! [Li + A − B − 1]q ! [A − i − n − 1]q ! [B + i − n]q ! [B]q ! 15) as special, respectively limiting cases. This determinant evaluation found applications in basic hypergeometric functions theory. In [191, Sec. 3], Wilson used a special case to construct biorthogonal rational functions. On the other hand, Schlosser applied it in [157] to find several new summation theorems for multidimensional basic hypergeometric series. In fact, as Joris Van der Jeugt pointed out to me, there is a generalization of Theorem 28 of the following form (which can be also proved by means of Lemma 5).

A proof of the first evaluation which uses Lemma 5 can be found in [155, Ch. VI, §3]. A proof of the second evaluation can be established analogously. Again, the second evaluation was always (implicitly) known to people in group representation theory, as it also results from a principal specialization (set xi = q i, i = 1, 2, . . ) of an odd orthogonal character of arbitrary shape, by comparing the orthogonal dual Jacobi–Trudi identity with the bideterminantal form (Weyl character formula) of the orthogonal character (cf.

2x + i + 3)! (3x + 2i + 5)i (3x + 2i + 8)i (x + 2i)! (x + 2i + 3)! × i=0 (2 n/2 − 1)!! 44) and for m = 4, n ≥ 4, it equals n/2 −1 n−1 i=0 i! (2x + i + 4)! (3x + 2i + 6)i (3x + 2i + 10)i (x + 2i)! (x + 2i + 4)! × 1 · (x + 1)(x + 2) (2x + 2i + 5) i=0 (2 n/2 − 1)!! (x2 + (4n + 3)x + 2(n2 + 4n + 1)) n even (2x + n + 4)(2x + 2n + 4) n odd. 45) One of the most embarrassing failures of “identification of factors,” respectively of LU-factorization, is the problem of q-enumeration of totally symmetric plane partitions, as stated for example in [164, p.

### Calculating some determinants by Krattenthaler C.

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