Determinants and Their Applications in Mathematical Physics
186 5. Further Determinant Theory =− [ 2 n− 1 ∑ i=1 (−1) i+1 Pf (n) i ai,^2 n ] 2 n− 1 ∑ j=1 (−1) j+1 Pf (n) j φj =−PfnH ...
5.3 The Matsuno Identities 187 B 2 n(φ)=−Hn− 1 H 3 n , B 2 n(φ, φ)=H 2 n− 1 H 2 n . (5.2.30) These identities arose by a by-prod ...
188 5. Further Determinant Theory and uij= 1 xi−xj =−uji, (5.3.2) where thexiare distinct but otherwise arbitrary. Illustration. ...
5.3 The Matsuno Identities 189 Hence, An=xAn− 1. ButA 2 =x 2 . The theorem follows. 5.3.2 Particular Identities............... ...
190 5. Further Determinant Theory where aij= { uij,j=i x− p ′′ n(xi) 2 p′n(xi) ,j=i. (5.3.7) This An clearly has the same value ...
5.3 The Matsuno Identities 191 Hence, if aij= { uij,j=i x− xi 1 −x 2 i ,j=i, then An=|aij|n=x n . (5.3.13) Exercises 1.LetAnden ...
192 5. Further Determinant Theory 5.4 The Cofactors of the Matsuno Determinant 5.4.1 Introduction Let En=|eij|n, where eij= { 1 ...
5.4 The Cofactors of the Matsuno Determinant 193 ( E ii + ∂ ∂xi ) E pru,qsv =e ipru,iqsv , (5.4.4) etc. 5.4.2 First Cofactors... ...
194 5. Further Determinant Theory m=1: ∑ r ∑ s † E is E rj =(ci−cj)E ij , which is equivalent to ∑ r ∑ s E is E rj − ∑ r E ir E ...
5.4 The Cofactors of the Matsuno Determinant 195 =−(ci−cj)E ij Equation (5.4.15) can be proved in a similar manner by appling (5 ...
196 5. Further Determinant Theory ∑ r,s † crcsE rs =− ∑ r,s † (c 2 r+c 2 s)E rs =− 1 3 ∑ r,s,t E rst,rst , (5.4.25) ∑ r,s crcsE ...
5.4 The Cofactors of the Matsuno Determinant 197 = ∑ i E ii ∑ r,s (cr+cs)E rs =2 ∑ i E ii ∑ r crE rr , (5.4.30) G=2 ∑ i,j,r crE ...
198 5. Further Determinant Theory = ∑ i,j,r E ijr,ijr , which is equivalent to (5.4.23). The application of a suitably modified ...
5.4 The Cofactors of the Matsuno Determinant 199 = ∑ r E rr ∑ i,j (c 2 i+c 2 j)E ij + ∑ r ∂ ∂xr ∑ i,j (c 2 i+c 2 j)E ij = ∑ r ( ...
200 5. Further Determinant Theory =U 2 −V 2 + ∑ r ∂S ∂xr , (5.4.42) where S= ∑ i,j cicjE ij . (5.4.43) This function is identica ...
5.5 Determinants Associated with a Continued Fraction 201 =4 ∑ r,s crcsE rs,rs +2 ∑ r,s c 2 sE rs,rs − 2 3 ∑ r,s,t,u E rstu,rstu ...
202 5. Further Determinant Theory = a 1 a 2 a 3 +a 1 b 3 +a 3 b 2 a 1 a 2 a 3 +a 1 b 3 +a 3 b 2 +a 2 a 3 b 1 +b 1 b 3 . Each of ...
5.5 Determinants Associated with a Continued Fraction 203 where Kn= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 b 1 − 1 a 1 b 2 − 1 a 2 b 3 − ...
204 5. Further Determinant Theory = Qn Pn , (5.5.8) wherePnandQneach satisfy the recurrence relation Rn=Rn− 1 +anxRn− 2 (5.5.9) ...
5.5 Determinants Associated with a Continued Fraction 205 etc. These formulas lead to the following theorem. Theorem 5.10. fn−fn ...
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