Page 329 - Determinants and Their Applications in Mathematical Physics

314 Appendix
A.4 Appell Polynomials

Appell polynomials φ m (x) may be defined by means of the generating
function relation
∞
φ m (x)t m
e G(t)=
xt
m!
m=0
∞ m−1
mφ m−1 (x)t
= , (A.4.1)
m!
m=1
where
∞
α r t r
G(t)= . (A.4.2)
r!
r=0
Differentiating the first line of (A.4.1) with respect to x and dividing the
result by t,
∞ m−1

φ (x)t
e G(t)= m
xt
m!
m=0
∞ m−1
φ 0 φ (x)t

= + m . (A.4.3)
t m!
m=1
Comparing the last relation with the second line of (A.4.1), it is seen that
φ 0 = constant, (A.4.4)
φ = mφ m−1 , (A.4.5)
m
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,
∞ ∞ ∞
φ m (x)t m α r t r (xt) m−r
=
m! r! (m − r)!
m=0 r=0 m=r(→0)
∞ ∞(→m)
t m m
= α r x m−r .
m! r
m=0 r=0
Hence,
m
m
φ m (x)= α r x m−r
r
r=0
m
m
= α m−r x ,
r
r
r=0
φ m (0) = α m . (A.4.6)