Determinants and Their Applications in Mathematical Physics

5.1 Determinants Which Represent Particular Polynomials 173

φn=

1

(1−x)

n+1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

1 −xx

11 −x 0

1 /2! 1 1 −x 0

1 /3! 1 /2! 1 1 −x 0

.....................................................

1 /(n−1)! 1/(n−2)!.............. 11 −x 0

1 /n!1/(n−1)!.............. 1 /2! 1 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

n+1

(5.1.8)

After expanding the determinant by the single nonzero element in the last

column, the theorem follows from (5.1.2) and (5.1.4).

Exercises

Prove that

1.

n

∑

r=0

αrx

n−r

y

r

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

α 0 α 1 α 2 α 3 ··· αn− 1 αn

−yx

−yx

−yx

.................

−yx

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

.

2.(x+y)

n

=

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1 xx

2

x

3

··· x

n− 1

x

n

− 1 yxyx

2

y ··· x

n− 2

yx

n− 1

y

− 1 yxy··· x

n− 3

yx

n− 2

y

− 1 y ··· x

n− 4

yx

n− 3

y

...........................

− 1 y

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

.

3.(−b)

n

2 F 0

(

x

a

,−n;−

b

a

)

=

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

−c 1 b

a −c 2 b

2 a −c 3 b

3 a −c 4

··· ··· ···

−cn− 1 b

(n−1)a −cn

∣

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

,

where

cr=(r−1)a+b+x. (Frost and Sackfield)

and 2 F 0 is the generalized hypergeometric function.