Difference Equations

and

Recursive Relations

1851-1875

Main Page History Topics Index


Many contributions have been made to the theory of difference equations and recursive equations by various mathematicians throughout the history of mathematics.  During the period of 1851 to 1875 some of the major contributors were:  Heine, Casorati, and Riemann.

Heinrich Eduard Heine worked on Legendre polynomials, Lamé functions and Bessel functions.

Heine-Sommerfeld functions are defined in terms of Bessel functions by the equations

Yn(z) = (p/2z)1/2 Jn+1/2(z),

zn(1)(z) = (p/2z)1/2 H(1)n+1/2(z),     zn(2)(z) =  (p/2z)1/2 H(2)n+1/2(z).

He is best remembered for the Heine-Borel theorem which stated that a subset of the reals is compact if and only if it is closed and bounded. Heine also formulated the concept of uniform continuity.
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Felice Casorati, an Italian mathematician, recognized many of the analogous properties of difference and differential equations. The Casoratian is the difference equation analog to the Wronksian in differential equations.  He developed the Casorati formula for linear difference equations.

Using N+ or Nn0+ as definition set , denoting a sequence by { yn }, which is the set of all values of the function y on  Nn0+ .

Definition 1
Let  p0 (n) = 1, p1 (n), ..., pk (n), gn be k + 2 functions defined on Nn0+An equation of the form

yn+k +  p1 (n) yn+k-1 + ... +  pk (n) yn = gn

(1)
is called a linear difference equation of order k, provided that pk (n) with a barperpendicular to the =  0.

With (1), the following k initial conditions

yn0 =  c1 ,   yn0+1 = c2 , ..., yn0+k-1 = ck

are associated where ci are real or complex constants.

Definition 2
If for every n e Nn0+ , g(n) = 0, then the equation (1) is said to be homogenous.

Definition 3
The function  fi (n), i= 1,2,...,k are linearly independent if for all n,
 

¥
å
n = 1
ai fi (n) = 0

implies ai = 0, i = 1,2,...,k.

A sufficient condition for the functions  fi (n), i= 1,2,...,k, to be linearly independent is that there exists an N two bars under the> n0 such that det K(N) with a barperpendicular to the = 0.

Let us now take k functions fi (n) defined on Nn0+ and define the matrix
 


 
 

K(n) = 

æ
ç
è
f1 (n)

f1 (n+1)

.  .  .

.  .  .

f1 (n+k-1)

 f2 (n)

f2 (n+1)

.  .  .

.  .  .

f2 (n+k-1)

.  .  .

.  .  .

.  .  .

.  .  .

.  .  .

 fk (n)

fk (n+1)

.  .  .

.  .  .

fk (n+k-1)

ö
÷
ø

Any other solution y(n, n0 , c) can be expressed as a linear combination of

y(n,  n0 , Ei ), i = 1, 2, ..., k.

The matrix K(n) is called the matrix of Casorati  (Casoratian) and it plays in the theory of difference equations the same role as the Wronskian (Wronski) matrix in the theory of linear differential equations.


Georg Friedrich Bernhard Riemann is known for using Riemann's zeta function to investigate the distribution of the prime numbers.

The Riemann zeta function z(s) is defined by
 

z(s) = 
¥
å 
n = 1
n-s ,

(s > 1).

z(2n) is a rational multiple of  p2n :
 

z(2n) =
22n-1 B
(2n)!
p2n

where Bn is Bernoulli's number.  Thus
 

z(s) =
p2
6
z(4) =
p4
90

One of the most striking property of the zeta function, discovered by Riemann himself, is the functional equation :
 

z(s) = c(s)z(1-s), c(s) = 2s ps-1 sin
æ
ç
è
ps
2
ö
÷
ø
G(1-s).
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The G(s) function is the Gamma function.

From the continuation of z(s) in the half plane Â(s) > 0, notice that the functional equation is
gives the analytic continuation of z(s) to the whole complex plane.

This analytic continuation can be obtained in several ways.

The complement formula of the Gamma function entails the formula:

c(s)c(1-s) = 1,

which gives a symetry of the functional equation with respect to the line Â(s) = 1/2.

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Article by:Sarah Pierson