Determinants and Their Applications in Mathematical Physics

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

∣

∣

n

, (4.9.5)

where

Cj=

[

φjφj+1φj+2···φj+n− 1

]T

,

φ

′

m=mF φm−^1.

Theorem 4.34.

T

′

=rF

∣

∣C

r− 1 Cr+1Cr+2···Cr+n− 1

∣

∣.

Proof.

C

′

j=F

(

jCj− 1 +C

∗

j

)

,

where

C

∗

j

=

[

0 φj 2 φj+1 3 φj+2···(n−1)φj+n− 2

]T

.

Hence,

T

′

=

r+n− 1

∑

j=r

∣

∣

CrCr+1···Cj− 1 C

′

j···Cr+n−^1

∣

∣

=F

r+n− 1

∑

j=r

∣

∣

CrCr+1···Cj− 1 (jCj− 1 +C

∗

j)···Cr+n−^1

∣

∣

=rF

∣

∣C

r− 1 Cr+1Cr+2···Cr+n− 1

∣

∣

+F

r+n− 1

∑

j=r

∣

∣

CrCr+1···C

∗

j···Cr+n−^1

∣

∣

after discarding determinants with two identical columns. The sum is zero

by Theorem 3.1 in Section 3.1 on cyclic dislocations and generalizations.

The theorem follows.

The column parameters in the above definition ofTare consecutive. If

they are not consecutive, the notation

Tj

1 j 2 ...jn

=

∣

∣C

j 1 Cj 2 ···Cjr···Cjn

∣

∣ (4.9.6)

is convenient.

T

′

j 1 j 2 ...jn

=F

n

∑

r=1

jr

∣

∣C

j 1 Cj 2 ···C(jr−1)···Cjn

∣

∣. (4.9.7)

Higher derivatives may be found by repeated application of this formula,

but no simple formula forD
k
(Tj
1 j 2 ...jn
) has been found. However, the