: Clarification Needed on Falling/Rising Factorial Notation in Arb Library & Documentation

[Stephen Crowley]

In the Arb library's documentation, the rising factorial function is denoted as `(x)_n`, a notation traditionally associated with the falling factorial. However, the function described computes the rising factorial, which is generally denoted differently, often as `x^(n)` or `x^(n)` with a bar over `n`.

This seems to contrast with standard mathematical notation, where `(x)_n` represents the falling factorial, defined as `x(x-1)(x-2)...(x-n+1)`, and `x^(n)` represents the rising factorial, defined as `x(x+1)(x+2)...(x+n-1)`.

Could you please clarify this notation in the Arb library's documentation? Is this a deliberate choice for a specific reason, or might it be an oversight?

Thank you for your time and attention to this matter. Your efforts in maintaining and improving the Arb library are greatly appreciated.

Best regards,

Stephen

[Stephen Crowley]

I understand it's just a matter of convention.. hypergeometric functions are defined in terms of rising factorials (hence the plus sign in prod(x+a_k,k=1..p) and that its just written counterintuitively as (x)_n instead of (x)^(n) as one might expect..

chatgpt seems to indicate thats right.. The rising factorial, often denoted as (x)n​, indeed represents a product of increasing terms starting from x and is a key component in the definition of hypergeometric functions. It's crucial to distinguish this notation from powers of x, which would typically be denoted as xn.

The notation (x)n​ for the rising factorial is standardized in mathematics to avoid confusion with exponentiation. It's defined as:

This notation is particularly prevalent in the study of hypergeometric series and special functions. In hypergeometric functions like p​Fq​, these rising factorials appear in the series expansion as a way to generalize the factorial function to non-integer values, providing a powerful tool in mathematical analysis and its applications.

[Jorge Zuniga]

Stephen,

This is a matter of convention.  In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol (x)n is used to represent the rising factorial. I think this is the notation followed in Arb and other Math softwares (Wolfram, Maplesoft, ...). On the other hand, Combinatorics uses a convention like you mentioned, but there is no such standard. It depends on the field you are involved in.

Jorge 

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[Stephen Crowley]

Thank you Jorge. Works like a charm :)

https://github.com/crowlogic/arb4j/blob/11d16027834826620d793b3792360758834c294c/src/main/java/arb/expressions/Expression.java#L764

public static void testRisingFactorial()

{

RealFunction func = RealFunction.express("x₍₃₎");

Real result = func.evaluate(new Real("5",

128),

0,

128,

new Real());

assertEquals(210.0, result.doubleValue());

}