Determinants and Their Applications in Mathematical Physics
106 4. Particular Determinants Let An=|φi+j− 2 |n=|φm|n, 0 ≤m≤ 2 n− 2 , Bn=|x i+j− 2 φi+j− 2 |n=|x m φm|n, 0 ≤m≤ 2 n− 2. (4.8.9) ...
4.8 Hankelians 1 107 The second proof illustrates the equivalence of row and column op- erations on the one hand and matrix-type ...
108 4. Particular Determinants 4.8.3 Two Kinds of Homogeneity.............. The definitions of a function which is homogeneous i ...
4.8 Hankelians 1 109 4.8.5 Turanians........................ A Hankelian in whichaij=φi+j−2+ris called a Turanian by Karlin and ...
110 4. Particular Determinants Proof. The identity is a particular case of Jacobi variant (A) (Sec- tion 3.6.3), ∣ ∣ ∣ ∣ ∣ T (n) ...
4.8 Hankelians 1 111 4.8.6 Partial Derivatives with Respect toφm....... InAn, the elementsφm,φ 2 n− 2 −m,0≤m≤n−2, each appear in ...
112 4. Particular Determinants The lemma follows from the second-order and third-order Jacobi identi- ties. 4.8.7 Double-Sum R ...
4.8 Hankelians 1 113 2 n− 2 ∑ m=1 mφm ∑ p+q=m+1 A ip A jq =(i+j−2)A ij . (D 2 ) These can be proved by puttingaij=φi+j− 2 andfr= ...
114 4. Particular Determinants Proof of (F).Denote the sum bySand apply the Hankelian relation φr+s− 3 =ar,s− 1 =ar− 1 ,s. S= ∑n ...
4.9 Hankelians 2 115 4.9 Hankelians 2 4.9.1 The Derivatives of Hankelians with Appell Elements The Appell polynomial φm= m ∑ r=0 ...
116 4. Particular Determinants wherePmis the Legendre polynomial, thenφmsatisfies (4.9.3) with F=(1−x 2 ) − 3 / 2 andφ 0 =P 0 = ...
4.9 Hankelians 2 117 2.The Yamazaki–Hori determinantAnis defined as follows: An=|φm|n, 0 ≤m≤ 2 n− 2 , where φm= 1 m+1 [ p 2 (x 2 ...
118 4. Particular Determinants 5.If An=|φm|n, 0 ≤m≤ 2 n− 2 , Fn=|φm|n, 1 ≤m≤ 2 n− 1 , Gn=|φm|n, 2 ≤m≤ 2 n, whereφm is an Appell ...
4.9 Hankelians 2 119 4.9.2 The Derivatives of Turanians with Appell and Other Elements Let T=T (n,r) = ∣ ∣C rCr+1Cr+2···Cr+n− 1 ...
120 4. Particular Determinants method can be illustrated adequately by taking the particular case in which (n, r)=(4,3) andφmis ...
4.9 Hankelians 2 121 Proof. The sum formula forTcan be expressed in the form n ∑ j=1 ψr+i+j− 1 T (n,r) ij =−δinT (n+1,r) n+1,n , ...
122 4. Particular Determinants Theorem 4.36. D(T (n,r) 11 )=−(2n+r−1)T (n,r+2) n,n− 1 . Proof. T (n,r) 11 =T (n− 1 ,r+2) . The t ...
4.10 Henkelians 3 123 Prove that D r (E)=(−1) r+1 r! n ∑ i=2 Si− 2 Eir =(−1) r+1 r! ∣ ∣C 1 C 2 ···Cr− 1 KCr+1···Cn ∣ ∣ n 4.10 He ...
124 4. Particular Determinants Identities 1. Vnr= n ∑ j=1 K rj n,^1 ≤r≤n. (4.10.4) Vnr= (−1) n+r (h+r+n−1)! (h+r−1)!(r−1)!(n−r)! ...
4.10 Henkelians 3 125 be expressed in the form Vnr= n ∏ j=1 (h+r+j−1) n ∏ i=1 i=r (r−i) − 1 = (h+r)(h+r−1)···(h+r+n ...
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