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Summation of Functions

The simplest type of equations in the theory of difference equations are difference equations of the type

Underscript[Δ, ω] F(x) Overscript[=, def] (F(x + ω) - F(x))/ω = φ(x) ,(1)

Underscript[∇, ω] G(x) Overscript[=, def] (G(x + ω) + G(x))/2 = φ(x),(2)

with the increment typeset structure being an arbitrary but fixed positive real number and with real or possibly complex typeset structure. The operator typeset structure is the so called first (Nörlund) difference quotient.  If typeset structure, we simply write typeset structure and typeset structure.

It is interesting to note that

FormBox[RowBox[{Underscript[lim, ω -> ∞] Underscript[Δ, ω] φ(x), , =, , RowBox[{d/(d x), Cell[], φ(x) .}]}], TraditionalForm]

The solutions of  (1) are called sums of typeset structure, and those of  (2) alternating sums of typeset structure, resp. The so-called general solution of   (1)  or of (2) can be represented as a sum of a particular solution and a solution of equation typeset structure, or typeset structure, resp. The central problem is to determine the particular solution by selection of its properties. Such particular solutions are then called principal sums. The condition are sometimes vague, and are of type:  

It can be easily verified that

A - ω Underoverscript[∑, j = 0, arg3] φ(x + j ω)      ... sp;     A + ω Underoverscript[∑, j = 0, arg3] φ(x - j ω)(3)

formally solves equation (1) with an arbitrary constant typeset structure, and that

2 Underoverscript[∑, j = 0, arg3] (-1)^j φ(x + j ω)     &n ... bsp;     - 2 Underoverscript[∑, j = 0, arg3] (-1)^j φ(x - j ω)(4)

formally solves (2). If these series are convergent (with typeset structure) they are usually taken for principal sums. For instance,

Nörlund [1] writes constant typeset structure is the form typeset structure and thus equation (3) gets the form

Underoverscript[∫, c, arg3] φ(x) d x - ω Underoverscript[∑, j = 0, arg3] φ(x + j ω)

with an arbitrary constant typeset structure.

To ensure the convergence of the involved series and the integral ingenious summation tricks are necessary in general to guarantee expected form of the principal sums. The general scheme of Nörlund’s approach can be describes as follows:

Suppose firstly that typeset structure is positive and continuous (complex or real) function for typeset structure. Then repalce the function typeset structure on the right hand side by another function  typeset structure with unknown typeset structure and depending on a new parameter typeset structure, which is chosen so that

a) typeset structure,

b) both typeset structure and typeset structure converge.

Then the function

F(x | ω, η) = Underoverscript[∫, c, arg3] φ(x, η) d x - ω Underoverscript[∑, s = 0, arg3] φ(x + s ω, η)

solves the difference equation

Underscript[Δ, ω] F(x) = φ(x, η),(5)

and if we let typeset structure, the relation (5) reduces to the equation and the function

F(x | ω) = Underscript[lim, η -> 0] F(x | ω, η)

is the solution of the equation (1) yields the principal sums provided the limit exists uniformly and is independent of the choice typeset structure  subject to the conditions a) and b) above.

An analogical scheme can also be applied to the equation (2).

Nörlund introduced a special notation to denote the principal sum of equation (1)

F(x | ω) = Overscript[Underscript[S, c], x] φ(z) Underscript[Δ, ω] z

and for the principal alternating sum of equation (2) he wrote

G(x | ω) = S φ(z) Underscript[∇, ω] z .

Nörlund applied the above scheme with  

φ(x, η) = φ(x)^(-ηλ(x)),

where typeset structure for some typeset structure, typeset structure.

Sufficient conditions under which the above requirements are fulfilled are [1] , pp.48-49:

(I) typeset structure has for typeset structure a continuous derivative of order typeset structure for some typeset structure such that typeset structure, and moreover

(IIa) typeset structure is uniformly comvergent in the interval typeset structure, and consequently in every interval typeset structure for any arbitrarily large typeset structure,

(IIb)  typeset structure is uniformly comvergent in the interval typeset structure (and consequently in every interval typeset structure for any arbitrarily large typeset structure).

Here the condition (I) and (IIa) are applied when solving the equation (2), that is in the case of the principal alternating sums, and (I) with (IIb) in case of principal sums.

The principal sums satisfy the following multiplication theorem (relation) [1] , p.44:

Underoverscript[∑, s = 0, arg3] F(x + (s ω)/m | ω) = m F(x | ω/m)(6)

while for the principal alternating sums of equation (2) there holds

Equation 2 there holds

Underoverscript[∑, s = 0, arg3] (-1)^s F(x + (s ω)/m | ω) = -ω/2 F(x | ω/m),(7)

Underoverscript[∑, s = 0, arg3] (-1)^s G(x + (s ω)/m | ω) = G(x | ω/m) .(8)

In addition we also have

G(x | ω) = 2/ω[F(x | ω) - F(x | 2 ω)] .(9)

Example. For typeset structure, typeset structure we get  [1] , p.53:

Overscript[Underscript[S, c], x] r z^(r - 1) Δ z = B _ r(z),      and          S z^r ∇ z = E _ r(z)

where typeset structure, and typeset structure denotes the typeset structureth Bernoulli, and Euler polynomial, resp. More generally

Overscript[Underscript[S, c], x] Underscript[r z^(r - 1) Δ, ω] z = ω^r B _ r(x/ω) .

The relations (7) and (8) contain the well-known multiplicative realtions for bernoulli and Euler polynomials discovered by Raabe in 1848 and the relation (9) the known relation bewteen Bernoulli and Euler polynomilas.

References

[1]  Nörlund, N. E. (1924). Vorlesungen über die Differenzenrechnung. Berlin: Springer Verlag.

Cite this web-page as:

Štefan Porubský: Summation of Functions.

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