Determinant identities

Determinants in Plane Partition Enumeration

“…[4]) have been found. By an interchange of the roles played by several variables in the WZ method, Petkovsek and Wilf [12] were able to find a genuinely short WZ proof of Theorem 2.…”

Quasi-L p norm orthogonal Galerkin expansions in sums of Jacobi polynomials

“…+ n 2 (48 − 66 μ 2 α 3 + 88 α − 72 μ − 72 μ α + 48 μ β + 124 μ β α + 32 μ β 2 α − 46 μ 2 α − 8 μ 2 α 4 + μ 2 β 3 α + 48 α 2 − 11 μ 2 α 3 β + μ 2 β 3 α 2 − 72 μ 2 β + 12 α 4 μ − 36 μ 2 β 2 + μ 2 α 5 + 20 α 3 μ β + 3 μ 2 β 2 α 3 + 56 α 3 μ + 96 α 2 μ β − 35 μ 2 β 2 α − 2 μ 2 β 2 α 2 + 44 α 2 μ + 16 μ 2 − 117 μ 2 α 2 + 3 μ 2 α 4 β + 24 μ β 2 −101 μ 2 α 2 β − 159 μ 2 α β + 8 α 3 + 8 μ β 2 α 2 ) +2 n 3 μ (α + β + 1) 2 α 3 μ + 8 α 2 − α 2 μ + 3 α 2 μ β + 7 μ β α − 29 μ α + 32 α + μ β 2 α − 30 μ + 2 μ β 2 + 4 μ β + 24 + n 4 μ 13 α 2 μ β + 24 + 37 μ β α + 7 α 3 μ + 8 α 2 + 6 μ β 2 α + 32 α + 12 μ β 2 − 4 μ α + 19 α 2 μ − 20 μ + 24 μ β + 6 n 5 μ 2 (α + 2) (α + β + 1) + 2 n 6 μ 2 (α + 2) p (2) 4 (n) = 4 (−1 + μ) 2 (α + 3) (α + 2) (α + 1) − nμ (2 + α + β) × 4 α 2 μ − 4 α 2 − 16 α + 16 μ α − μ β − 12 + 12 μ + n 2 μ 12 + μ β 2 α + 2 α 2 μ β + 8 μ β α + 4 α 2 + 2 μ β 2 + 16 α − 4 μ α + 2 α 2 μ − 4 μ + α 3 μ + 9 μ β + 2 n 3 μ 2 (α + 2) (2 + α + β) + n 4 μ 2 (α + 2)…”

“…This is easily illustrated by combining Chebyshev polynomials of the first kind, T n (x), of various degrees to fit the boundary condition f (1) + f (1) = 0. Two possible sets of basis functions are {ψ (1) n } and {ψ (2) n }, each formally spanning the same function space, where ψ (1) n (x) = T n (x) − (n 2 + 1)T 0 (x), ψ (2) n (x) = (n − 1) 2 + 1 T n (x) − (n 2 + 1)T n−1 (x). We note that ψ (1) n becomes increasingly ill-conditioned as n increases, as when normalized, ψ (1) n (x) ∼ T 0 , which is independent of n. The second case, ψ (2) n , is better conditioned but is neither an orthogonal set, nor close to equalripple.…”

“…Two possible sets of basis functions are {ψ (1) n } and {ψ (2) n }, each formally spanning the same function space, where ψ (1) n (x) = T n (x) − (n 2 + 1)T 0 (x), ψ (2) n (x) = (n − 1) 2 + 1 T n (x) − (n 2 + 1)T n−1 (x). We note that ψ (1) n becomes increasingly ill-conditioned as n increases, as when normalized, ψ (1) n (x) ∼ T 0 , which is independent of n. The second case, ψ (2) n , is better conditioned but is neither an orthogonal set, nor close to equalripple. An alternative approach is to use a Gram-Schmidt procedure [5] with the boundary condition to generate an orthogonal set, by writing ψ (3) n = n i=0 a n P (α,β) n (x), where the a i are determined from imposing orthogonality (with weight function w(x) = (1 − x) α (1 + x) β ) to ψ (3) i , i = 1, .…”

“…, n − 1, the boundary condition and some normalization condition. 2 Although guaranteed to produce an orthogonal set, this algorithm is cumbersome for large n and provides no means to prove any asymptotic, or other, properties of the functions.…”