First let me recapitulate the method from my point of view.

We start with as usual with a function and want to obtain a superfunction of it, i.e. a function that satisfies

(1) .

Now by differentiating once we get

expanding twice:

and generally by induction:

making our new variable:

To extend the product to non-integer we have a look at the sum

(2)

Extending the sum to non-integer boundaries

The sum can extended to non-integer values via Faulhaber's formula which provides the sum for monomials:

,

where is the -th Bernoulli polynomial with the Bernoulli numbers.

The sum operator is linear for natural boundaries so one can extend Faulhaber's formula to linear combinations of monomials (also called polynomials ), and finally to powerseries. So for non-integer we define for

the extended sum .

To simplify the matter we can define the antidifference or indefinite sum:

and get

Extended sum and shifts

We hope that the extended sum behaves like a normal sum with integer boundaries, e.g. we want to have

(3) .

We show it for polynomials. Let be a polynomial of degree , then we know that (3) is satisfied for all integers and both sides are polynomials of maximal degree in . But equality at different points already implies the equality of both polynomials.

Hence not only for integer but for every the equality holds.

This than carries over to powerseries (as limit of a polynomial sequence) if no convergence issues arise.

This property lets us formulate (2) in a much nicer way by applying (3) with and afterwards setting :

(4)

Uniqueness of extended sum superfunction

Sketch of proof:

Let beside be another at analytic super-exponential (1) that satisfies (2). We further assume and to be invertible at and . We set in a vicinity of , so there.

and by (1) so for integer .

So put into (4), with :

the left difference can be replaced with (4), knowing :

(5)

Now consider the forward difference

.

Hence

for any up to perhaps a constant, so

.

Now we apply this to (5):

together with and this gives and , so and .

We start with as usual with a function and want to obtain a superfunction of it, i.e. a function that satisfies

(1) .

Now by differentiating once we get

expanding twice:

and generally by induction:

making our new variable:

To extend the product to non-integer we have a look at the sum

(2)

Extending the sum to non-integer boundaries

The sum can extended to non-integer values via Faulhaber's formula which provides the sum for monomials:

,

where is the -th Bernoulli polynomial with the Bernoulli numbers.

The sum operator is linear for natural boundaries so one can extend Faulhaber's formula to linear combinations of monomials (also called polynomials ), and finally to powerseries. So for non-integer we define for

the extended sum .

To simplify the matter we can define the antidifference or indefinite sum:

and get

Extended sum and shifts

We hope that the extended sum behaves like a normal sum with integer boundaries, e.g. we want to have

(3) .

We show it for polynomials. Let be a polynomial of degree , then we know that (3) is satisfied for all integers and both sides are polynomials of maximal degree in . But equality at different points already implies the equality of both polynomials.

Hence not only for integer but for every the equality holds.

This than carries over to powerseries (as limit of a polynomial sequence) if no convergence issues arise.

This property lets us formulate (2) in a much nicer way by applying (3) with and afterwards setting :

(4)

Uniqueness of extended sum superfunction

Sketch of proof:

Let beside be another at analytic super-exponential (1) that satisfies (2). We further assume and to be invertible at and . We set in a vicinity of , so there.

and by (1) so for integer .

So put into (4), with :

the left difference can be replaced with (4), knowing :

(5)

Now consider the forward difference

.

Hence

for any up to perhaps a constant, so

.

Now we apply this to (5):

together with and this gives and , so and .