Determinants and Their Applications in Mathematical Physics

314 Appendix

A.4 Appell Polynomials

Appell polynomialsφm(x) may be defined by means of the generating

function relation

e

xt

G(t)=

∞

∑

m=0

φm(x)t

m

m!

=

∞

∑

m=1

mφm− 1 (x)t

m− 1

m!

, (A.4.1)

where

G(t)=

∞

∑

r=0

αrt

r

r!

. (A.4.2)

Differentiating the first line of (A.4.1) with respect toxand dividing the

result byt,

e

xt

G(t)=

∞

∑

m=0

φ

′

m

(x)t

m− 1

m!

=

φ

′

0

t

+

∞

∑

m=1

φ

′

m

(x)t

m− 1

m!

. (A.4.3)

Comparing the last relation with the second line of (A.4.1), it is seen that

φ 0 = constant, (A.4.4)

φ

′

m

=mφm− 1 , (A.4.5)

which is a differential–difference equation known as the Appell equation.

Substituting (A.4.2) into the first line of (A.4.1) and using the upper and

lower limit notation introduced in Appendix A.1,

∞

∑

m=0

φm(x)t

m

m!

=

∞

∑

r=0

αrt

r

r!

∞

∑

m=r(→0)

(xt)

m−r

(m−r)!

=

∞

∑

m=0

t

m

m!

∞(→m)

∑

r=0

(

m

r

)

αrx

m−r

.

Hence,

φm(x)=

m

∑

r=0

(

m

r

)

αrx

m−r

=

m

∑

r=0

(

m

r

)

αm−rx

r

,

φm(0) =αm. (A.4.6)