Determinants and Their Applications in Mathematical Physics

A.10 Formulas Related to the Function (x+

√

1+x^2 )

2 n

333

Proof. Replacexby−x
− 1
in (A.10.1), multiply byx
2 n
, and putx
2
=z.

The result is

(−1+

√

1+z)

2 n

=

[

z

n

+

n

∑

i=1

λniz

n−i

]

−(1 +z)

1

2

n

∑

i=1

μniz

n−i

=

n+1

∑

j=1

λn,n−j+1z

j− 1

−(1 +z)

1

2

n

∑

j=1

μn,n−j+1z

j− 1

.

Rearrange, multiply by (1 +z)

− 1 / 2

and apply (A.10.4):

∞
∑

i=0

νiz

i

n+1

∑

j=1

λn,n−j+1z

j− 1

=

n

∑

j=1

μn,n−j+1z

j− 1

+(1+z)

− 1 / 2

(−1+

√

1+z)

2 n

.

In some detail,

(1 +ν 1 z+ν 2 z

2

+···)(λnn+λn,n− 1 z+···+λn 1 z

n− 1

+λn 0 z

n

)

=(μnn+μn,n− 1 z+···+μn 1 z

n− 1

)+

(

z

2

) 2 n

(1 +z)

− 1 / 2

(

1 −

1

4

z+···

) 2 n

.

Note that there are no terms containingz
n
,z
n+1
,...,z
2 n− 1
on the right-

hand side and that the coefficient ofz
2 n
is 2
− 2 n

. Hence, equating coefficients

ofz
n−1+i
,1≤i≤n+1,

n+1

∑

j=1

λn,j− 1 νi+j− 2 =

{

0 , 1 ≤i≤n

2

− 2 n

,i=n+1.

The theorem appears when n is replaced by (n−1) and is ap-

plied in Section 4.11.3 in connection with a determinant with binomial

elements.

It is convenient to redefine the functionsλnrandμnrfor an application

in Section 4.13.1, which, in turn, is applied in Section 6.10.5 on the Einstein

and Ernst equations.

Ifnis a positive integer,

(x+

√

1+x

2

)

2 n

=gn+hn

√

1+x

2

, (A.10.6)

where

gn=

n

∑

r=0

λnr(2x)

2 r

, an even function,

g 0 =1;

hn(x)=

n

∑

r=1

μnr(2x)

2 r− 1

, an odd function,

h 0 =0. (A.10.7)