Determinants and Their Applications in Mathematical Physics

320 Appendix

The polynomial

ψmn(x)=

m

∑

r=0

(

m

r

)

αn+rx

r

satisfies the relations

ψ

′

mn=mψm−^1 ,n+1,

ψmn−ψm− 1 ,n=xψ

′

mn

=mxψm− 1 ,n+1.

Exercises

1.Prove that

φm(x−h)=

m

∑

r=0

(

m

r

)

(−h)

r

φm−r(x)

=∆

m

hφ^0.

2.If

Sm(x)=

∑

r+s=m

φrφs,

Tm(x)=

∑

r+s+t=m

φrφsφt,

prove that

S

′

m

=(m+1)Sm− 1 ,

Sm(x+h)=

m

∑

r=0

(

m+1

r

)

h

r

Sm−r(x),

T

′

m=(m+2)Tm−^1 ,

Tm(x+h)=

m

∑

r=0

(

m+2

r

)

h

r

Tm−r(x).

3.Prove that

φ

− 1

m =

1

αm

∞

∑

n=0

(−1)

n

cmnx

n

,

where

cm 0 =1,

cmn=

1

α

n

m

∣

∣

∣

∣

(

m

m−i+j− 1

)

αm−i+j− 1

∣

∣

∣

∣

n

,n≥ 1.

This determinant is of Hessenberg form, is symmetric about its sec-

ondary diagonal, and contains no more than (m+ 1) nonzero diagonals

parallel to and including the principal diagonal.