C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54--102.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....b)b (a Gamma b 1) a b) A combination of these results and use of the identity (see [19, p. 463, 133) 2 F 1 1; 1 3 2 ; z = arcsin p z p z(1 Gamma z) finish the proof. 3. Sketch of proof of Lemma 6 The method that we use for this proof is also applied successfully in [12, 9, 10, 11, 13] (see in particular the tutorial description in [11, Sec. 2] First of all, we take appropriate factors out of the determinant in (2.7) To be precise, we take (N m Gamma i) m i Gamma 1) 2N Gamma 2i 1) out of the i th row of the determinant, i = 1; 2; N . Thus we obtain N ....

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54--102.

.... j 1) 2, B j = m Gamma j 1. After some simplification one arrives at the right hand side of (5.2) The determinant in (4.3) evaluates as follows. 1 (2m i 1) i : 5.3) Proof. The method that we use for this proof is also applied successfully in [10, 7, 8, 9, 11] (see in particular the tutorial description in [9, Sec. 2] First of all, as in the proof of Lemma 9, we take appropriate factors out of the determinant. To be precise, we take (2n m Gamma i Gamma 1) out of the i th row of the determinant in (5.3) i = 1; 2; 2n Gamma 1. Thus ....

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54--102.

....of factorials and binomials (cf. 10, x5.5, 5.96) 5.100) Theorem 1 would also make sense and be true for complex x and y. Proof. We prove the determinant evaluation by identi cation of factors, a method that is also applied successfully in [2] 3] 5] 6] 7] 11] 12] 13] 14] [16], 17] and [21] see in particular the tutorial description in [15, x2.4] or [13, x2] First of all, we take appropriate factors out of the determinant. To be precise, we take (x y j) x n 2j) y n j) out of the j th column of the determinant in (2) j = 1; 2; n. Thus we ....

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54-102.

....determine this constant in the third step, computation of the multiplicative constant, one compares coefficients of x on both sides of (2.19) This is an enjoyable exercise. Consult [91] if you do not want to do it yourself. Further successful applications of this procedure can be found in [27, 30, 42, 89, 90, 92, 94, 97, 132]. Having done that, let me point out that most of the individual steps in this sort of calculation can be done (almost) automatically. In detail, what did we do We had to 1. Guess the result. Indeed, without the result we could not have got started. 2. Guess the vectors in the kernel. 3. ....

....settled, the authors of [16] expect important implications in the approximation of algebraic numbers. The case N = 1 of Theorem 49 is a special case of (3.11) and, thus, on a shallow level. On the other hand, the next case, N = 2, is already on a considerably deeper level. It was first proved in [94], by establishing, in fact, a much more general result, given in the next theorem. It reduces to the N = 2 case of Theorem 49 for x = 0, b = 4l, and c = 2l. In fact, the x = 0 case of Theorem 50 had already been conjectured Theorem 50. Let b; c be nonnegative integers, c b, and let Delta(x; b; ....

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C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54--102. (p. 15, 45, 45, 45, 45, 54)

....of factorials and binomials (cf. 10, x5.5, 5.96) 5.100) Theorem 1 would also make sense and be true for complex x and y. Proof. We prove the determinant evaluation by identi cation of factors, a method that is also applied successfully in [2] 3] 5] 6] 7] 11] 12] 13] 14] [16], 17] and [21] see in particular the tutorial description in [15, x2.4] or [13, x2] First of all, we take appropriate factors out of the determinant. To be precise, we take (x y j) x n 2j) y n j) out of the j th column of the determinant in (2) j = 1; 2; n. Thus ....

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54-102.

.... computation of the multiplicative constant, one compares coefficients of x ( n 2 ) on ADVANCED DETERMINANT CALCULUS 15 both sides of (2.19) This is an enjoyable exercise. Consult [91] if you do not want to do it yourself. Further successful applications of this procedure can be found in [27, 30, 42, 89, 90, 92, 94, 97, 132]. Having done that, let me point out that most of the individual steps in this sort of calculation can be done (almost) automatically. In detail, what did we do We had to 1. Guess the result. Indeed, without the result we could not have got started. 2. Guess the vectors in the kernel. 3. ....

....important implications in the approximation of algebraic numbers. The case N = 1 of Theorem 49 is a special case of (3.11) and, thus, on a shallow level. On the other hand, the next case, N = 2, is already on a considerably deeper ADVANCED DETERMINANT CALCULUS 45 level. It was first proved in [94], by establishing, in fact, a much more general result, given in the next theorem. It reduces to the N = 2 case of Theorem 49 for x = 0, b = 4l, and c = 2l. In fact, the x = 0 case of Theorem 50 had already been conjectured in [16] Theorem 50. Let b; c be nonnegative integers, c b, and let ....

[Article contains additional citation context not shown here]

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54--102. (p. 15, 45, 45, 45, 45, 54)

.... b) A combination of these results and use of the identity (see [17, p. 463, 133) 2 F 1 1; 1 3 2 ; z = arcsin p z p z(1 Gamma z) finish the proof. THE NUMBER OF RHOMBUS TILINGS 9 3. Proof of Lemma 5 The method that we use for this proof is also applied successfully in [13, 2, 10, 11, 12, 14] (see in particular the tutorial description in [12, Sec. 2] First of all, we take appropriate factors out of the determinant in (2.7) To be precise, we take (N m Gamma i) m i Gamma 1) 2N Gamma 2i 1) out of the i th row of the determinant, i = 1; 2; N . Thus we obtain N ....

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math., 3 (1997), 54--102.

....(2n m Gamma i Gamma 1) m i Gamma 1) 4n Gamma 2i Gamma 1) 2n Gamma2 Y i=1 (2m i 1) i Theta n Gamma1 X i=0 ( Gamma1) n Gammai Gamma1 (2n Gamma 2i Gamma 1) m n Gamma i) 2i i 2 : 5.3) Proof. The method that we use for this proof is also applied successfully in [10, 7, 8, 9, 11] (see in particular the tutorial description in [9, Sec. 2] First of all, as in the proof of Lemma 15, we take appropriate factors out of the determinant. To be precise, we take (2n m Gamma i Gamma 1) Gamma (m i Gamma 1) 4n Gamma 2i Gamma 1) Delta out of the i th row of the ....

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, New York J. Math. 3 (1997), 54--102.

....(2n m Gamma i Gamma 1) m i Gamma 1) 4n Gamma 2i Gamma 1) 2n Gamma2 Y i=1 (2m i 1) i Theta n Gamma1 X i=0 ( Gamma1) n Gammai Gamma1 (2n Gamma 2i Gamma 1) m n Gamma i) 2i i 2 : 5.3) Proof. The method that we use for this proof is also applied successfully in [8, 5, 6, 7, 9] (see in particular the tutorial description in [7, Sec. 2] First of all, as in the proof of Lemma 15, we take appropriate factors out of the determinant. To be precise, we take (2n m Gamma i Gamma 1) Gamma (m i Gamma 1) 4n Gamma 2i Gamma 1) Delta out of the i th row of the ....

C. Krattenthaler and D. Zeilberger, Proof of a determinant evaluation conjectured by Bombieri, Hunt and van der Poorten, preprint.

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