Determinants and Their Applications in Mathematical Physics
166 4. Particular Determinants β 2 i, 2 i=λii,i≥ 1 , β 2 i+1, 2 j+1=λij, 0 ≤j≤i, β 2 i+2, 2 j=λi+1,j−λij, 1 ≤j≤i+1, (4.13.6) λij ...
4.13 Hankelians 6 167 the upper limits that γij= i ∑ r=1 j ∑ s=1 βir 2 r+s− 1 kr+s− 2 βjs. Hence, γ 2 p+1, 2 q+1=2 2 p+1 ∑ r=1 2 ...
168 4. Particular Determinants Define three other matricesM ′ ,K ′ , andN ′ of ordernas follows: M ′ =[α ′ ij]n (symmetric), K ′ ...
4.14 Casoratians — A Brief Note 169 Qn= 1 2 |qij|n, (4.13.19) where pij=t|i−j|−ti+j, qij=t|i−j|+ti+j− 2 , (4.13.20) appear in Se ...
5 Further Determinant Theory 5.1 Determinants Which Represent Particular Polynomials 5.1.1 Appell Polynomial................... ...
5.1 Determinants Which Represent Particular Polynomials 171 b.ψn(x)= 1 n! ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ α 0 α 1 α 2 α 3 ··· αn− 1 αn nx ...
172 5. Further Determinant Theory 5.1.2 The Generalized Geometric Series and Eulerian Polynomials Notes on the generalized geome ...
5.1 Determinants Which Represent Particular Polynomials 173 φn= 1 (1−x) n+1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 1 −xx 11 −x 0 1 /2! 1 1 −x ...
174 5. Further Determinant Theory 5.1.3 Orthogonal Polynomials Determinants which represent orthogonal polynomials (Appendix A.5 ...
5.1 Determinants Which Represent Particular Polynomials 175 But a ′ ij +ai− 1 ,j= [( j− 1 i− 1 ) + ( j− 1 i− 2 )] u (j−i+1) − [( ...
176 5. Further Determinant Theory =D n ( yA 1 v ) , ( A 1 =u= vy ′ y ) =D n+1 (y). Hence, An+1= v n+1 D n+1 (y) y , which is equ ...
5.1 Determinants Which Represent Particular Polynomials 177 2.Hn(x)= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 x 2 12 x 4 12 x 6 12 x 8 ··· ·· ...
178 5. Further Determinant Theory 6.Prove that the determinantAnin (5.1.10) satisfies the relation An+1=vA ′ n +(u−nv ′ )An. Put ...
5.2 The Generalized Cusick Identities 179 In particular, ai, 2 n= 2 n−i ∑ s=1 φs+i− 1 ψ 2 n−s, 1 ≤i≤ 2 n− 1. (5.2.4) LetA 2 nden ...
180 5. Further Determinant Theory Assume that Pfm=HmKm,m<n. (5.2.10) The method by which the theorem is proved for all values ...
5.2 The Generalized Cusick Identities 181 s 2 =ai+1,j+φiψj, s 3 =ai+1,j+1. The lemma follows. Let A ∗ 2 n =|a ∗ ij | 2 n, Pf ∗ ...
182 5. Further Determinant Theory LetH ∗ n− 1 andK ∗ n− 1 denote the determinants obtained fromHn− 1 and Kn− 1 , respectively, b ...
5.2 The Generalized Cusick Identities 183 = n ∑ j=1 n+j− 1 ∑ i=j H (n) jnK (n) i−j+1,nx 2 n−i− 1 = 2 n− 1 ∑ i=1 x 2 n−i− 1 n ∑ j ...
184 5. Further Determinant Theory =Hn n+1−j ∑ s=1 ψ 2 n−sδs,n−j+1 =Hnψn+j− 1 , 1 ≤j≤n; (5.2.21) Yi= n ∑ j=1 K (n) jn 2 n−i−j+1 ∑ ...
5.2 The Generalized Cusick Identities 185 From (5.2.4), ∂ai, 2 n ∂φ 2 n− 1 =ψi. Also, ∂Hn ∂φ 2 n− 1 =Hn− 1. Hence, 2 n− 1 ∑ i=1 ...
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