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Difference Operators

Difference equation is an equation involving differences between values of a function often (but not necessarily) of  a discrete variable. Difference equations are special types of functional equations which are equations (or systems of equations) in which a function - usually as a functions of a continuous variable - is sought to satisfy certain relations among its values at all points. For example, Cauchy determined  [1] , (2) 3, p.98, p.220,  the general continuous solutions of each of the simple functional equations (the so-called Cauchy functional equations): , , , . On the other hand, Gamma function satisfies the difference equation .

There are at least three interpretations of a difference equation:

• as a sequence of numbers,
• as a discrete dynamical system,
• as an iterated function.

A difference operator is an operator which maps a function, say , to another one of the type , where are given parameters. This operator plays in the calculus of finite differences formally similar role to that of the derivative.

Basic difference operators are [2] , [3]:

• , the so-called first (Nörlund) difference quotient  (sometimes also discrete-time delta operator),
• 
  with the increment being an arbitrary but fixed positive real number and with real or possibly complex ,
• If , we simply write for .  The operator is called the forward difference operator,
• and analogously defined operator   is called the backward difference operator,
• If the increment of the variable is not constant, then the so-called divided differences are taken into account. If are distinct vales of the argument then the first divided difference is defined as  .  Clearly, .

We can continue and define the second order difference operators, for instance the second divided difference as



etc. The th forward difference if a function is given by



where is the binomial coefficient.

Since

and due to the properties of difference operators listed below,  the difference equations can often be solved with techniques very similar to those used for solving differential equations.

The following basic rules govern the manipulation with forward and backward difference operators:

• if is a constant then , ,
• if are constants, then , the so called linearity,
• product rules: , ,   
• quotient rule: ,  and  ,
• summation rules: ,  .

References

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