Determinants and Their Applications in Mathematical Physics
206 5. Further Determinant Theory a 3 = |c 0 | ∣ ∣ ∣ ∣ c 1 c 2 c 2 c 3 ∣ ∣ ∣ ∣ |c 1 | ∣ ∣ ∣ ∣ c 0 c 1 c 1 c 2 ∣ ∣ ∣ ∣ , (5.5.19) ...
5.5 Determinants Associated with a Continued Fraction 207 Proof. Let f 2 n− 1 P 2 n− 1 −Q 2 n− 1 = ∞ ∑ r=0 hnrx r , (5.5.23) whe ...
208 5. Further Determinant Theory The equation hn, 2 n+1=0 yields n+1 ∑ t=0 c 2 n+1−tp 2 n+1,t=0. (5.5.29) Applying the recurren ...
5.5 Determinants Associated with a Continued Fraction 209 5.5.3 Further Determinantal Formulas Theorem 5.12. a.P 2 n− 1 = 1 An ∣ ...
210 5. Further Determinant Theory b.Q 2 n= 1 Bn ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ c 1 c 2 c 3 ··· cn+1 c 2 c 3 c 4 ··· cn+2 .................... ...
5.6 Distinct Matrices with Nondistinct Determinants 211 = 1 Bn r ∑ t=0 cr−tB (n+1) n+1,n+1−t. Hence, from the fourth equation in ...
212 5. Further Determinant Theory determinants with constant elements. It is a trivial exercise to find two determinantsA=|aij|n ...
5.6 Distinct Matrices with Nondistinct Determinants 213 Define a TuranianTnr(Section 4.9.2) as follows: Tnr= ∣ ∣ ∣ ∣ ∣ ∣ ∣ φr− 2 ...
214 5. Further Determinant Theory coaxial in the sense that all their secondary diagonals lie along the same diagonal parallel t ...
5.6 Distinct Matrices with Nondistinct Determinants 215 |α 2 |=− ∣ ∣ ∣ ∣ ∣ ∣ 12 x 1 xx 2 φ 0 φ 1 φ 2 ∣ ∣ ∣ ∣ ∣ ∣ =− ∣ ∣ ∣ ∣ ∣ ∣ ...
216 5. Further Determinant Theory 1 − 4 x 10 x 2 1 − 3 x 6 x 2 − 10 x 2 1 − 2 x 3 x 2 − 4 x 3 5 x 4 1 −xx 2 −x 3 x 4 −x 5 1 α 0 ...
5.6 Distinct Matrices with Nondistinct Determinants 217 ∣ ∣ ∣ ∣ φ 1 (x+y) φ 2 (x+y) φ 2 (x+y) φ 3 (x+y) ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ ∣ 1 ...
218 5. Further Determinant Theory Once again the symbol←→ denotes that the columns are arranged in reverse order. Illustrations ...
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5.7 The One-Variable Hirota Operator 221 5.7 The One-Variable Hirota Operator 5.7.1 Definition and Taylor Relations Several nonl ...
222 5. Further Determinant Theory Applying Taylor’s theorem again, 1 2 {φ(x+z)−φ(x−z)}= ∞ ∑ n=0 z 2 n+1 D 2 n+1 (φ) (2n+ 1)! , 1 ...
5.7 The One-Variable Hirota Operator 223 Proof. First proof(Caudrey). The Hessenbergian satisfies the recurrence relation (Secti ...
224 5. Further Determinant Theory = ∞ ∑ m=0 z m um+1 m! ∞ ∑ r=0 z r Fr r! . (5.7.13) Equating coefficients ofz n , Fn+1 n! = n ∑ ...
5.7 The One-Variable Hirota Operator 225 =D r {F(x)+(−1) r G(x)},D= d dx . (5.7.18) Hence, ψ 2 r=D 2 r log(fg) =D 2 r (φ) =u 2 r ...
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