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can't think of a single practically useful univariate generating function that has bad analytic properties. By "practically useful", I mean something like "can be found printed in a paper or book". Obviously any precise definition will say almost no sequences of reals have convergent generating functions, but that's just not interesting. The multivariate case is another story, particularly with infinitely many variables, but that's not what the person was talking about.
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Wait, what's that? Nor does the rest of the article really make clear what "the general linear recurrence problem" is. It talks about finding a closed-form solution given a recurrence relation, and about extracting a recurrence relation given a generating function. Is "the" general linear recurrence
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Additionally, I am a bit biased towards the content in the wiki and it is hard for me to point out precise areas which might prove to be educationally ill-formed to most. So I would like some feedback in that direction, thank you! (Ex: The 'Article is a mess' post above seems rather insightful, and
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Finite sequences embed into infinite sequences in a natural way, by appending all 0s. So, for example, the sequence of coefficients of the series you mention can be understood to be (0, 0, ..., 0, (2/5)^64, ..., 1/5^64, 0, 0, 0, ...). The emphasis on "infinite" in the lead is slightly misplaced.
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The writing frequently feels inappropriate for an encyclopedia. It's often clearly trying to teach the reader from the ground up rather than summarize the topic, like in "Example 3: Generating functions for mutually recursive sequences". Consequently it's often long-winded with frequent asides and
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Also, how about this one: We just list a couple applications of generating functions (I honestly think snake oil or something is a good enough thing to convince people that they're 'useful', and then maintain a 'main article' on applications). I wish to scrap off the entire J-fraction part, write
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of the article indicates that "generating series" is "more correct" than "generating function." While I agree that generating functions aren't really functions (for instance, because their evaluation at specific points isn't what they're about), I worry that they aren't really series either (for
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Very old thread, but I don't agree. A huge number of interesting generating functions are meromorphic. The main heuristic motivation for using exponential generating functions is often that the coefficients grow too quickly for an ordinary generating function to converge. Off the top of my head I
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By far the biggest issue: the material on OGF's and EGF's needs to be split into its own articles. This article should be a panoramic view of generating functions with tons of links to specific instances (as is already done for
Lambert, Bell, and formal Dirichlet series). The current version is
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Good point about uniqueness of two-sided inverses, probably worth saying in the article. Instead of saying this is unique, say this is a two-sided inverse, and thus it is unique. (I'm not sure if two-sided inverses are unique in non-associative rings, but I think that's out-of-scope for the
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There are tons of "local" issues, like the fact that none of the "precise, technical" definitions actually reference base rings or power series, the large number of lengthy equations that should be displayed rather than in-line, the ad-hoc, inconsistent use of theorem-like "environments"....
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The information here is really not enough... it didn't give me any idea how to calculate the generating function coefficients. It's algebra and series, but the article should list the most used tricks: binomial theorem, infinite geometric series, convolution products, etc.
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The first sentence of the introduction says a generating function is "an infinite sequence of numbers". The second sentence says it is a single number, namely: "the sum of this infinite series". Apart from the morph of "sequence" into "series", this is pretty confusing.
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instance, because whether or not they converge isn't what they're about). Given that there is now a citation (which I haven't checked!) to show that "generating series" is also in use, might we simply say that it is an "alternative" rather than "more correct"?
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the sum; this sum is not a number because the individual summands are not numbers. I think Bill's suggestion is a completely acceptable alternative for that sentence. The immediately following sentence leaves something to be desired, as well.
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Spreading the content to multiple articles that specialize in specific aspects sounds good to me. However, much of the content is quite interesting to me, so I am hoping that, as it leaves this article, it does go somewhere, not merely to the
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It seems to be a complaint that the article is too huge to read. I was wondering if we can cut some sections down. Obviously there must have been those before me who wondered, so I mean to ask: What's a systematic way to maintain such a list?
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I said the growth rate is "often" the main heuristic. It's certainly not the only one. I also did not say there are no "useful" univariate generating functions with bad analytic properties, though I like the examples in your paper, like
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This article is fairly typical of current
Knowledge (XXG) mathematics articles: it dives headlong into a mass of detail without first explaining the basics. This is supposed to be an online encyclopedia, not a maths textbook!
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For starters, we should probably remove P-holonomic functions and J-fractions and give them their dedicated pages. But beyond that, at the time of writing this, I am not sure of what optimisations one can perform.
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1936:{\displaystyle \pi (\cot(\pi (c+z))-2\cot(2\pi (c-z))-2\cot(2\pi (c+z))+\cot(\pi (c-z)))=-2\left(\sum _{k=0}^{\infty }z^{2k}\sum _{x=1}^{\infty }{\frac {1}{(x+c-1/2)^{2k+1}}}-{\frac {1}{(x-c-1/2)^{2k+1}}}\right)}
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Indeed, it is very, very old. But as long as it is being revived: you are wrong about both the heuristic and the substance. The "right" heuristic for exponential generating functions is about labelings, and
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is a practically useful (in your definition) use of generating functions with bad analytic properties (lazily drawn from my own work because only one example is necessary to make the point). See also
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I noticed that all summation formulae on the page look like this: for each natural n sum a_i*x^n or something. I believe this should be fixed, because 1/(1-x) is not x+x^2+x^3+... but 1+x+x^2+x^3...
2625:. Please be open to making an improvement given that the inadequacy of the current version has been pointed out by more than one person. I.e., please come up with a version that is better than both
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That's a nice extension of the given statement. All the article actually uses is that formal power series with coefficients in any ring form a ring -- two-sided inverses are unique in any ring. --
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I would avoid the use of either "sum" or "summation" in this setting. I agree with Joel's objection to using "summation" and I am also not happy with the original phrasing. I would suggest using,
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The first sentence was not very clearly written, I have tried to rephrase it (in keeping also with the general rule that encyclopedia articles are about things, not about names for things). --
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In hindsight, the applications part can be cut down here and there. However, it's the ordinary generating functions part that needs to basically go out of the window. It's WAY too extensive.
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section with four examples of different types of generating function for the sequence of square numbers, and also an extended example showing how the ordinary generating function for the
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We could start by checking that all the content on ordinary generating functions makes it to an ordinary generating functions wiki, and then we can prune most of the content from here.
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some irrelevant bits, like "We suggest an approach by generating functions." Every word should be carefully weighed to decide if it's worth saying, which by no means has been done.
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I am not an expert on the field, so I will not dare to introduce the following definition myself. But if somebody does agree, please include under "Definitions" the following:
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Ah, sorry about that. On the first one I should have said that you kept it while changing the adjacent sentence after RobLandau flagged the wording of both sentences. Sorry.
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and friends. My point was to respond to `To call these things "continuous" is absurd', when it's frequently not, especially in the context of an introduction to the subject.
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2206:{\displaystyle \pi ^{3}(8(\cot(2c\pi )+\cot ^{3}(2c\pi ))-(\cot(c\pi )+\cot ^{3}(c\pi )))=\sum _{x=1}^{\infty }{\frac {1}{(x+c-1/2)^{3}}}-{\frac {1}{(x-c-1/2)^{3}}}}
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something about them in a main link, write about transforming between ordinary and exponential generating functions and then remove the whole transforming part.
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In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers.
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too much at once and mainly succeeds in doing many things badly. The length is probably dissuading people from wanting to jump in and help clean up as well.
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Your action that I was referring to was your revert of my edit at 11:48 today, which restored what I had altered, and not your earlier edit on 8 February.
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I made a change to the article, dropping a condition (something being an integral domain) on the explanation of the uniqueness of the inverse of (1-
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with the text "power series", then drops in the phrase "formal power series" without explaining what "formal" means in this context, then links to
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Well, I don't think that's very clear. The powers of a variable are really place-olders, here. There is no necessary connection to continuity.
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Recently someone asked for the probability distribution of the sum of 64 rolls of a biased die, and I replied by expanding the polynomial
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I made an edit to the article. If that's not right somehow, please fix or revert it, and/or continue the discussion here. Thank you —
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Loraof, I do not think your description of my actions is accurate: the unique edit I made in response to RobLandau's comments here is
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This article has so many issues. I'll list the biggest ones in the hope that (perhaps over years) they'll eventually get fixed.
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2484:{\displaystyle \cot(\pi (c+z))\approx \cot(c\pi )-z\pi (1+\cot ^{2}(c\pi ))+z^{2}\pi ^{2}(\cot(c\pi )+\cot ^{3}(c\pi ))}
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243:(1994), sections 5.2 and 5.3. If you wanted to check those and insert any that are applicable, that'd be great! —
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I agree. Many useful generating functions are not continuous or even convergent. Any definition must stress the
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Education is a process of diminishing deception. Start off with the simple stuff; the ifs and buts come later.
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I just noticed that the german version is not liked here it's called "Erzeugende
Funktionen", url is here:
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on
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Two of us, RobLandau and myself, have now pointed out that the status quo ante of this sentence, which
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is any ring with a unit, not necessarily commutative or an integral domain, then the only power series
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Please give the references for the very nice formulas in the section on asymptotics of coefficients.
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have an identity element 1. Multiplication by -1, and its necessary properties, is then implied by
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The sum (that is, the whole infinite series) is the GF. "Summation" does not make sense here.)
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Please help fix the broken anchors. You can remove this template after fixing the problems. |
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are integers" is certainly a lot less jargony than using the blackboard bold notation. --
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taking multiple derivatives with respect to z closed form sums can be obtained such as:
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I find a paper that uses a formula quite like this and cites G. Pólya and G. Szegő,
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In this section we give formulas for generating functions enumerating the sequence
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In this section we give formulas for generating functions enumerating the sequence
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The following formula is really easy to use. Shall it be included in this article?
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I'll try to propose concrete edits which might circumvent the proposed issues.)
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the article would be more difficult to follow, as you would not know what the
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Absolutely. (No pun intended.) To call these things "continuous" is absurd.
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sequences. The article does not say that; the article says they are used to
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is the octonions, for example.) It is only required that multiplication in
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problem just the challenge of understanding linear recurrences in general?
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has twice restored, doesn’t make sense. The original and restored sentence
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Generating functions were first introduced by
Abraham de Moivre in 1730
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This article links to one or more target anchors that no longer exist.
969:{\displaystyle (1-X)f(X)=f_{0}+(f_{1}-f_{0})X+(f_{2}-f_{1})X^{2}+...}
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Your comment describes me has having "twice restored" something. --
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Then the generating function for $ s_n$ is given by the formula
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be a linear recursive sequence of order k with initial conditions
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2972:. Is that not a generating function because it's not infinite? —
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Formula for generating function for a linear recursive sequrnce.
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The summation of this infinite series is the generating function
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being an integral domain is required. Indeed multiplication in
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has been used, and commutativity of this operation , that is,
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An editor has asked for a discussion to address the redirect
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The sum of this infinite series is the generating function
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in order to solve the general linear recurrence problem.
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need not even be associative. (So, the result holds if
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My replacement sentence, which I’m not wedded to, said
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3670:Done; thanks for your help ironing this out! --
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676:{\displaystyle 1=(1-X)f(X)}
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540:section you will find an
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3149:and recursive relation
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450:{\displaystyle \Psi (x)}
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397:, notes on Chapter 1. --
372:14:45, 15 May 2019 (UTC)
152:project's priority scale
2799:if you wish to do so.
2797:the redirect discussion
241:generatingfunctionology
109:WikiProject Mathematics
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1357:{\displaystyle a=0}
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1270:{\displaystyle F]}
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346:Michael Hardy
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148:High-priority
143:
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73:High‑priority
71:
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3472:MOS:STYLERET
3422:
3395:— Preceding
3233:
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549:
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511:
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485:— Preceding
479:
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184:Anchors are
181:
147:
107:
51:WikiProjects
34:
3743:null device
3432:this revert
2801:Jasper Deng
2554:Jasper Deng
2497:—Preceding
1435:—Preceding
1367:—Preceding
1026:means that
976:. Solving
554:Definitions
538:Definitions
528:84.9.78.198
123:Mathematics
114:mathematics
70:Mathematics
3789:Categories
3028:XOR'easter
2633:. Thanks!
1411:article.)
633:such that
283:Definition
3617:Quantling
3426:Quantling
3401:Kaiwang45
3021:— fine —
3004:submarine
2793:Function*
2631:summation
2603:summation
2540:RobLandau
2524:RobLandau
1555:Quantling
1530:Gandalf61
562:Gandalf61
39:is rated
3770:Yeetcode
3760:contribs
3752:uantling
3728:Yeetcode
3713:Yeetcode
3662:contribs
3654:uantling
3606:contribs
3598:uantling
3568:, where
3505:, where
3409:contribs
3397:unsigned
2676:this one
2550:describe
2499:unsigned
1437:unsigned
1429:Formulae
1394:Charleyc
1381:contribs
1369:unsigned
830:. Then
558:Examples
550:Examples
542:Examples
487:unsigned
258:contribs
250:uantling
3441:MOS:BBB
2974:Tamfang
2600:. Here
579:). If
201:before.
150:on the
41:C-class
3672:Beland
3635:Beland
3526:, and
3466:Beland
3450:Beland
2804:(talk)
2755:Loraof
2699:Loraof
2635:Loraof
2557:(talk)
329:formal
292:S = {a
222:Asympt
47:scale.
3581:<
3532:<
3448:. --
3015:again
2650:This
1490:with
1338:then
28:This
3774:talk
3756:talk
3732:talk
3717:talk
3676:talk
3658:talk
3639:talk
3627:and
3602:talk
3577:0 ≤
3575:and
3528:0 ≤
3454:talk
3405:talk
3051:Let
3032:talk
2994:talk
2978:talk
2839:talk
2759:talk
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2629:and
2570:talk
2528:talk
2507:talk
1578:talk
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1548:The
1534:talk
1445:talk
1417:talk
1413:DRLB
1398:talk
1377:talk
1373:DRLB
1218:nor
495:talk
463:talk
403:talk
391:here
368:talk
333:Zero
275:Iopq
254:talk
226:talk
182:Tip:
142:High
3745:. —
3573:≥ 2
3524:≥ 2
2990:JBL
2823:way
2725:JBL
2685:JBL
2627:sum
2619:sum
2566:JBL
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2433:cot
2380:cot
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2310:cot
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2241:sin
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2035:cot
1999:cot
1974:cot
1716:cot
1680:cot
1641:cot
1605:cot
1242:in
683:is
514:...
399:JBL
395:EC1
309:."
3791::
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3762:)
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3660:|
3641:)
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3604:|
3550:+
3548:an
3519:,
3514:∈
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3347:−
3336:∗
3325:−
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3209:−
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3115:…
3034:)
2996:)
2988:--
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2841:)
2761:)
2731:)
2705:)
2691:)
2683:--
2680:is
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2473:π
2464:
2445:π
2436:
2421:π
2398:π
2389:
2367:π
2361:−
2355:π
2346:
2340:≈
2319:π
2313:
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2225:−
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2008:
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1977:
1956:π
1893:−
1887:−
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1779:∞
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1719:
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1129:…
1104:−
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996:−
929:−
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818:…
731:…
653:−
603:∈
526:--
497:)
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436:Ψ
405:)
370:)
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256:|
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3772:(
3754:(
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439:(
401:(
366:(
307:S
302:n
300:a
296:}
294:n
273:-
252:(
247:Q
224:(
154:.
53::
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