Determinants and Their Applications in Mathematical Physics

4.7 Wronskians 97

and where

φ

′

m=(m+1)φm−^1 ,φ^0 = constant,

prove that

A

′

n

=n(n−1)An− 1.

3.Prove that

n

∏

r=1

∣

∣

∣

∣

1 arx

− 11

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

1 b 12 xb 13 x

2

··· ··· b 1 ,n+1x

n

− 11 b 23 x ··· ··· b 2 ,n+1x

n− 1

− 11 ··· ··· b 3 ,n+1x

n− 2

··· ··· ··· ···

− 11

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

n+1

,

where

bij=

j− 1

∏

r=i

ar.

4.If

Un=

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣

u

′

u

′′

/2! u

′′′

/3! u

(4)

/4! ···

uu

′

u

′′

/2! u

′′′

/3! ···

uu

′

u

′′

/2! ···

uu

′

···

··· ···

∣

∣

∣

∣ ∣ ∣ ∣ ∣ ∣ n

,

prove that

Un+1=u

′

Un−

uU

′

n

n+1

. (Burgmeier)

4.7 Wronskians

4.7.1 Introduction

Letyr=yr(x), 1≤r≤n, denotenfunctions each with derivatives of

orders up to (n−1). These functions are said to be linearly dependent if

there exist coefficientsλr, independent ofxand not all zero, such that

n

∑

r=1

λryr= 0 (4.7.1)

for all values ofx.

Theorem 4.24. The necessary condition that the functionsyrbe linearly

dependent is that

∣

∣

y

(i−1)

j

∣

∣

n

=0

identically.