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  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.