Determinants and Their Applications in Mathematical Physics

318 Appendix

If

φm=

(1 +x)

m+1

−cx

m+1

m+1

,

then

θm=

x

m+1

+(−1)

m

c

(m+ 1)(1 +x)

m+1

,

θ 0 =

x+c

1+x

.

The Taylor Series Solution

Functionsφm(x) which satisfy the Appell equation (A.4.5) but are not

Appell sets according to the strict definition given above may be called

Appell functions, but they should not be confused with the four Appell

hypergeometric series in two variables denoted byF 1 ,F 2 ,F 3 , andF 4 , which

are defined by Whittaker and Watson and by Erdelyi et al.

The most general Taylor series solution of (A.4.5) for givenφ 0 which is

valid in the neighborhood of the origin is expressible in the form

φm=

m

∑

r=1

(

m

r

)

αrx

m−r

+m

∫x

0

φ 0 (u)(x−u)

m− 1

du, m=1, 2 , 3 ,....

A proof is given by Vein and Dale. Hildebrand obtained a similar result by

means of the substitutionφm=m!fm, which reduces (A.4.5) to

f

′

m=fm−^1.

Multiparameter and Multivariable Appell Polynomials

The Appell equation (A.4.5) can be generalized in several ways. The two-

parameter equation

u

′

ij=iui−^1 ,j+jui,j−^1 (A.4.12)

is a differential partial difference equation whose general polynomial

solution is

uij(x)=

i

∑

r=0

j

∑

s=0

(

i

r

)(

j

s

)

αrsx

i+j−r−s

,i,j=0, 1 , 2 ,...,

where theαrsare arbitrary constants.

u 00 =α 00 ,

uij(0) =αij.

A proof can be constructed by applying the identity

i

(

i− 1

r

)(

j

s

)

+j

(

i

r

)(

j− 1

s

)

=(i+j−r−s)

(

i

r

)(

j

s

)

.