Maxieds

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GeoffreyT2000 (talk, contribs) 23:27, 24 March 2017 (UTC)

Copy-editing

I've done a number of edits to the article titled Generating function transformation and I may be back. Please note that:

• You should not assume at the outset that the reader knows that the article is about mathematics just because of its title. Non-mathematicans are likely not to know that "Generating function transformations" implies it's about mathematics. You can begin by saying "In mathematics, ..." or "In number theory, " or "In algebra, " etc. (but not "In category theory, ..." or "In topology, " because most non-mathematicians don't know what those are). Sometimes that is unnecessary because of the title of the article; e.g. if the title is "Mathematical induction", that that makes it clear.
• You used far too many capital letters in section headings. Look around at Wikipedia articles. One does not capitalize an initial letter merely because it's in a section heading. This is codified in WP:MOS.
• One uses en-dashes, not hyphens, in things like "Laplace–Borel" and in ranges of pages, e.g. pp. 37–48 or years, etc.
• Note this difference between \mathrm and \operatorname:

${\begin{aligned}&a\operatorname {sech} b&&\qquad {\text{This uses operatorname.}}\\&a\operatorname {sech} (b)&&\qquad {\text{This uses operatorname.}}\\&a\mathrm {sech} b&&\qquad {\text{This uses mathrm.}}\\&a\mathrm {sech} (b)&&\qquad {\text{This uses mathrm.}}\end{aligned}}$

\operatorname{} results in proper spacing to the left and right, and the contrast between the first two lines shows that the spacing depends on the context. \mathrm{} does not result in proper spacing.

Michael Hardy (talk) 04:39, 25 March 2017 (UTC)

And, issues aside, I like to thank Maxieds for his/her effort. Good job, anyway! Boris Tsirelson (talk) 07:49, 25 March 2017 (UTC)

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The link is now changed

I didn't mean for the link to point to the disambiguation page. The problem is that there is really no main page for Hadamard products of series besides the subsection linked in generating function transformations. I fixed the problem by removing the link entirely. Perhaps I should update the Hadamard product page to point to this reference for Hadamard products of generating functions and formal power series (and DONE). (Maxie)

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Generating function

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A miscommunication

Hi Maxieds,

I just wanted to clarify that my edit summary here was not intended as an imperative; rather, it was supposed to be a description of my own edit. (Possibly, the misunderstanding here is on my end -- my comment was prompted by this edit summary.)

All the best, JBL (talk) 02:49, 13 April 2017 (UTC)

It's ok, I didn't take the comment personally. Do you think that the updated identity is correct? Maxie (talk) 02:52, 13 April 2017 (UTC)

Yes, it looks right now. (Is there any reason not to cancel the redundant (n!)^m terms from both sides?) --JBL (talk) 18:03, 13 April 2017 (UTC)

Looking back at the previous identity I wanted to check yesterday, it looks almost correct given that the terms when k=1 (i.e., with $s1(n+1,1)\equiv n!$) are not getting indexed in the expansions. The (n!)^m terms are not entirely redundant on both sides of the equation since we are scaling the harmonic numbers to get integers on both sides unless you want to change this for some reason. This is a convention I used in forming the harmonic number sequences and generating functions in this article since it makes forming diagonal generating functions of the integer-valued Stirling numbers easier.

The idea of what I was looking for in the expansions on this page is to get terms (even though $s1(n+1,1)/n!=1$ in many of the terms) that preserve the "isobaric" nature of the indices in each term of the expansions. If you look carefully you will notice that the indices in each term all add up to $2m$, which doesn't happen if you start canceling out terms that are equal to one. The Mathematica code I used to check the identity from my scratch work is actually

Table[{Index -> m, Expand[-m SeriesCoefficient[Log[1 + Sum[s[k + 1]/s[1] Power[-x, k], {k, 1, 24}]], {x, 0, m}]*Power[s[1], m]]}, {m, 1, 5}] // TF

which requires some accounting for the factorials in the denominator of the generating function to get the isobaric structure of the coefficients. The above Mathematica code generates terms like $s[2]^{2}-2s[1]s[3]$ and $s[2]^{3}-3s[1]s[2]s[3]+3s[1]^{2}s[4]$ which respect this property. So in short, this identity is definitely trickier than the first exponential harmonic number generating function for the Stirling numbers. I hope that answers what you were asking. -- Maxie (talk) 18:28, 13 April 2017 (UTC)

Ok, thanks. --JBL (talk) 20:10, 21 April 2017 (UTC)

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Divisor sum identities

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Prime omega function

Hello.

I have a few comments about the new article titled prime omega function.

• The link to moment was to a disambiguation page that lists various songs titled "Moments" and magazines and movies bearing that title, and to Cheese Flavoured Moments and to a model of Android phone, the Samsung Moment, and a lot of other things, and that should be expected every time one links to a word that means various different things in different fields.
• The link to factorial moments didn't work because you used the plural, rather than factorial moment. You can write [[factorial moment]]s, and the reader sees factorial moments, and when you click on it, it goes to the page titled factorial moment. Article titles are singular except when there is a special reason to use the plural. I have since created a redirect page from "factorial moments" to "factorial moment".
• Note the use of \mid:

$7\mid 42\,$ is coded as 7\mid 42.

$7|42\,$ is coded as 7 | 42.

the first of these above is standard for "7 divides 42". Likewise

$7\nmid 42\,$ is coded as 7 \nmid 42.

• In 2017–2018, one uses an en-dash, not a hyphen.

right: 2017–2018

wrong: 2017-2018

Similarly in ranges of pages: pp. 406–423.

Michael Hardy (talk) 17:30, 25 April 2018 (UTC)

Hi Micheal, I could have sworn that we already had an article on the omega function, more than ten years ago. But I searched for it, I could not find it. I could have sworn that it already contained the info in prime omega function, and more. Am I imagining things? 67.198.37.16 (talk) 20:54, 25 January 2019 (UTC)

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Your October 17th edit to Liouville function: incomplete sentence

Your edit to Liouville function left an incomplete sentence in the opening. It currently says "expressed in terms of the Moebius function and the [[". Can you please complete your edit? Thanks! 134.114.109.109 (talk) 05:17, 29 November 2018 (UTC)

Done. Thanks for alerting me. -- Maxie — Preceding unsigned comment added by 128.61.125.81 (talk) 12:14, 29 November 2018 (UTC)

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Lambert Series

Hi Maxie, I was wondering if you knew of a reference for this factoid ... from a quick numeric exploration, its obvious that it has an asymptotic expansion

$\sum _{n=1}^{\infty }{\frac {q^{n-1}}{(1-q^{n})}}={\frac {-\log(1-q)}{1-q}}+{\frac {\gamma }{1-q}}-{\frac {1}{2}}\log(1-q)+{\mathcal {O}}(1-q)$

as $q\to 1$ and $\gamma$ the Euler-Mascheroni. I was wondering if you knew of any references for this, or general treatments of these limits? The above is probably not too hard to derive; I am being merely lazy. (the gamma and the one-half log are dead give-aways that this is "well-known" somehow.) Also, just glancing at the pretty picture that someone added to the Lambert series article makes it blaringly obvious that some modular symmetry is at work; that its some modular form. I presume that any totally multiplicative function would yield some modular form... (Perhaps its not actually modular, but somehow almost-so?) Would you know of any references for this? Would you care to expand the article on this? Thanks! 67.198.37.16 (talk) 20:51, 25 January 2019 (UTC)

Heh. One reason I ask is that there is variant of the Redheffer matrix, without the extra column of ones; let me call it the "divisor operator" (its a matrix with non-zero entries from divisibility). The divisor operator is not a bounded operator on $\ell _{1}$ and the above Lambert series shows up in the proof of unboundedness. Perhaps you are aware of other work in this Redheffer-variant? Does it have a name? Any clue why Redheffer tacked on the extra column of ones? (I actually have more questions, let me know if you want to hear them.) 67.198.37.16 (talk) 22:18, 25 January 2019 (UTC)

RE: talk (Lambert series)

Lambert series expansions

I do not actually know this asymptotic expansion nor where to find a reference for it. To be honest, my experience with Lambert series is almost exclusively treating them formally as [generating function]s for certain (multiplicative) number theoretic functions. The specific series you cited is a shifted OGF for the [divisor function], $d(n+1)\equiv \sigma _{0}(n+1)$. With respect to it being related to modular forms, you are probably right. Maybe you can check out some references to series expansions of Ramanujan's mock theta functions?

Redheffer matrices

Most of what I currently know about Redheffer matrices has already been added to the relevant Wikipedia page by my edits from several weeks ago (including some interpretations of these 0-1 matrices in terms of divisor sums). I do not know why Redheffer considered these matrices with the extra columns of ones. Perhaps some insight can be gleaned from | this paper in Section 5.1 where the breakdown of determinants into a sum of two matrices is defined.

RE: Further questions

Sure. Ask away. I hope I can be of some help. What is your username? You can email me if you'd prefer not to disclose your identity on this site (maxieds@gmail)...

-- Maxie (--Maxie (talk) 06:48, 28 January 2019 (UTC))

I can be reached at linasvepstas@gmail ... I've managed to forget whatever additional questions I might have had. I'm wildly over-committed on far too many projects, and so really should not be soliciting any additional conversations to take part in. I was just thinking of the divisor function as an infinite-dimensional matrix operator, discovered that it had a name: its a variant of a Redheffer matrix; I figured out that its not a bounded operator on the Banach l_1 space: the Lambert series above corresponds to the vector that "blows up". All the while, I was looking at the edit history of the various pages, observed that you had recently expanded them, and so I thought I would drop in and say "hello". So.. hello! 67.198.37.16 (talk) 06:12, 5 February 2019 (UTC)

Hello! I actually have some active research related to matrices of these general "Redheffer-like" forms as discussed in later sections of the original article. I have not yet encountered a treatment of these matrix types that treats them as (bounded) operators, say in a Banach space, but I do see quite a few immediate combinatorial interpretations and similar constructions of sums which can be phrased as matrix-vector multiplication problems involving these forms. I'd be happy to talk more about it over email if you come up with any other interesting related topics or questions. Maxie (talk) 20:53, 25 February 2019 (UTC)

Dirichlet series inversion moved to draftspace

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@Cabyi: Yes, I'm aware that the current page is lacking references right now. Please bear with me as I expand the page and gradually add the standard references to the identities I have cited ... — Preceding unsigned comment added by Maxieds (talk • contribs) 06:53, 24 September 2019 (UTC)

It's in draftspace now, you can take your time. There's no rush. Cabayi (talk) 07:22, 24 September 2019 (UTC)

While you're working on it, the first person pronouns need rewriting, WP:TONE. Hope that helps, Cabayi (talk) 08:33, 24 September 2019 (UTC)

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