Determinants and Their Applications in Mathematical Physics

A.6 The Generalized Geometric Series and Eulerian Polynomials 325

Lawden’s functionSm(x) is defined as follows:

Sm(x)=(1−x)

m+1

ψm(x),m≥ 0. (A.6.11)

It follows from (A.6.5) thatSmis a polynomial of degreemin (1−x) and

hence is also a polynomial of degreeminx. Lawden’s investigation into

the properties ofψmandSmarose from the application of thez-transform

to the solution of linear difference equations in the theory of sampling

servomechanisms.

The Eulerian polynomial Am(x), not to be confused with the Euler

polynomialEm(x), is defined as follows:

Am(x)=(1−x)

m+1

φm(x),m≥ 0 , (A.6.12)

Am(x)=Sm(x),m> 0 ,

A 0 (x)=1,

S 0 (x)=x, (A.6.13)

Am(x)=

m

∑

n=1

Amnx

n

, (A.6.14)

where the coefficientsAmnare the Eulerian numbers which are given by

the formula

Amn=

n− 1

∑

r=0

(−1)

r

(

m+1

r

)

(n−r)

m

,m≥ 0 ,n≥ 1 ,

=Am,m+1−n. (A.6.15)

These numbers satisfy the recurrence relation

Amn=(m−n+1)Am− 1 ,n− 1 +nAm− 1 ,n. (A.6.16)

The first few Eulerian polynomials are

A 1 (x)=S 1 (x)=x,

A 2 (x)=S 2 (x)=x+x

2

,

A 3 (x)=S 3 (x)=x+4x

2

+x

3

,

A 4 (x)=S 4 (x)=x+11x

2

+11x

3

+x

4

,

A 5 (x)=S 5 (x)=x+26x

2

+66x

3

+26x

4

+x

5

.

Smsatisfies the linear recurrence relation

(1−x)Sm=(−1)

m− 1

m− 1

∑

r=0

(−1)

r

(

m

r

)

(1−x)

m−r

Sr

and the generating function relation

V=

x(x−1)

x−e

u(x−1)

=

∞

∑

m=0

Sm(x)u

m

m!

,