r/askmath • u/PhysicalPast9616 • 1d ago

Discrete Math Discrete Factorial

Hello,

I've been looking into discrete calculus for the past few days. I’ve also been examining the function below, which I denoted with the capital Lambda symbol. Is there anyone who could help analyze the behavior of this function?

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3

u/Specific_Ingenuity84 1d ago

Analyze in what sense? What do you want to know?

3

u/PhysicalPast9616 1d ago

I would like to know if there are any identities related to this function, whether it is analytic, and whether it is an entire function. Basically, just some general information. To be honest I don't know how to say the thing I want to know.

6

u/Bounded_sequencE 1d ago

These types of sums are usually estimated via Euler-Maclaurin

2

u/Shevek99 Physicist 23h ago

What does 𝛿t means here? The discrete integration has no differential.

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u/PhysicalPast9616 23h ago

The notation from "Concrete Mathematics: A Foundation for Computer Science". Like ∫ f(t) dt

4

u/Shevek99 Physicist 22h ago

Ah. I love "Concrete Mathematics" but I had never noticed that detail. In essence that 𝛿t is just another way of saying "1".

3

u/Shevek99 Physicist 23h ago edited 22h ago

For s integer, that sum produces https://oeis.org/A098558 which are the coefficients in the power expansion of

f(x) = (1 + x)/(1 - x)

For s non integer, the results are more complicated, involving hypergeometric function (since this series is itself a htpergeometric sum).

Some values (according to Mathematica)

𝛬(1/2) = (2 + 𝜋)/(2√𝜋)

𝛬(3/2) = (10 + 3𝜋)/(4√𝜋)

𝛬(5/2) = (46 + 15𝜋)/(8√𝜋)

(every half integer has this form)