Wikipedia:Articles for deletion/Hyperbolic Fibonacci and Lucas functions

The following discussion is an archived debate of the proposed deletion of the article below. Please do not modify it. Subsequent comments should be made on the appropriate discussion page (such as the article's talk page or in a deletion review). No further edits should be made to this page.

The result was delete. The delete side has a substantial majority despite spirited defenses by WIKIWIZDOM and KennKramer, and looking at the arguments, it seems that both the numbers and the governing guidelines and policies support deletion.

The argument in the "delete" vote seems well grounded in policy. The article references extensively an article by Vera W. de Spinadel (for some reason linked to a top page, but the article in question is probably this one to cover the phyllotaxis sections, yet while that reference gives extensive coverage on Fibonacci numbers and metallic means, no mention at all is made of hyperbolic or Lucas functions. Stringing together different topics to give support to another topic is considered excellent practice in academia when this is done correctly, but on Wikipedia, this is considered original research by synthesis. Several of the delete votes have expressed concern that this part of the article constitutes a fringe theory.

The argument of notability is also relevant. Most academic articles provide results of some new research, that is after all the purpose of academic journals. However, the new theory that comes from each of these articles does not usually create a new and notable topic that is suitable for Wikipedia. Notability guidelines emphasize independent sourcing, which means that a new idea should be utilized or analyzed in a non-trivial manner (e.g. not mere citations) by independent sources before being considered as a topic for Wikipedia. The Russian references provided by Kmarinas86 are relevant but having looked at some of the material using Google Translate, it is far from clear that the references to Stakhov go beyond the citation level.

In response to the last note by KennKramer note that Wikipedia is a tertiary source that reflects what the current consensus is on the topics that we cover. KennKramer points out that "As a rule, all new math concepts initially met a resistance from the mathematical community.", however that is not a reason to include this particular topic on Wikipedia. As a philosophical matter, those concepts that met resistance initially but are accepted today are the ones that you see. What is less easy to see is the much larger number of novel ideas and concepts that met initial resistance and that never emerged from obscurity, and probably never will emerge from obscurity. Wikipedia cannot speculate on which of the currently obscure ideas will eventually emerge to become widely covered topics, nor can Wikipedia pass judgement about which of these ideas ought to emerge from obscurity.

With the numbers clearly in favor of deletion, and with those arguments solidly backed up by policy and guideline, there is a rough consensus in this debate for deleting. Sjakkalle (Check!) 14:53, 9 February 2013 (UTC)

Hyperbolic Fibonacci and Lucas functions

Hyperbolic Fibonacci and Lucas functions (edit | talk | history | protect | delete | links | watch | logs | views) – (View log · Stats)

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The subject of this article on mathematics is not notable. The mathematical content of the article consists in results that are trivial consequences of known theories, typically that of linear recurrences. When these known theories are cited in the article, this results always of my edits. Instead of referring to knowns theories, the article cites only non-notable publications that, for most of them, are not reliably published. The part of the article devoted to phylotaxis is a blatent WP:fringe theory and I suspect that it is also pseudo-science. See the talk page for more details D.Lazard (talk) 23:21, 24 January 2013 (UTC)

• Delete Per Lazard, basically. As far as I can see, the topic doesn't seem to be major enough for encyclopedic treatment: the Google search with ""Hyperbolic Fibonacci and Lucas functions" -stakhov" is very discouraging. It is often hard to show something not-notable. But, absent convincing counter-evidence, I have to go with "non-notable". -- Taku (talk) 23:51, 24 January 2013 (UTC)

Taku. Its unclear why your Google search with "Hyperbolic Fibonacci and Lucas functions" has led you to discouraging results. Google search, made by me, led me to find more than 30 articles on the hyperbolic Fibonacci and Lucas functions, published in reputable journals such as, Physics Letters, Chaos, Solitons and Fractals, Communications in Theoretical Physics, International Journal of Contemporary Mathematical Sciences, Applied Mathematics and Computation, Complex Geometry, Patterns, and Scaling in Nature and Society, International Journal of Physical Sciences, World Journal of Modelling and Simulation, Nonlinear Dynamics, Communications in Nonlinear Science and Numerical Simulation, Artificial Intelligence Conference Proceedings, Journal of Applied Mathematics, Journal of Mathematics, etc. All of these articles are listed in the section “Further reading”.WIKIWIZDOM (talk) 17:58, 25 January 2013 (UTC)

— WIKIWIZDOM (talk • contribs) has made few or no other edits outside this topic.

Notice. Independent source material is largely in the Russian language: It is highly probable that the attention on this mathematical subject is strongly Russian in origin. The Russian spelling for "the hyperbolic Fibonacci and Lucas functions" is "гиперболическими функциями Фибоначчи и Люка". "гиперболическими" is Russian for "hyperbolic". "функциями" is Russian for "functions". "Фибоначчи" is Russian "Fibonacci". "Люка" is Russian for "Lucas". The Russian spelling for the name "Stakhov" is "Стахов" or "Стахова" (if followed by "and".

2380 results for "Hyperbolic functions" Fibonacci: https://www.google.com/search?q=%22гиперболических+функций%22+Фибоначчи+

1700 results (%71 of total) for "Hyperbolic functions" Fibonacci -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22гиперболических+функций%22+Фибоначчи+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

1320 results for "Hyperbolic function Fibonacci": https://www.google.com/search?q=%22гиперболических+функций+Фибоначчи%22

9 results (<%1 of total) for "Hyperbolic function Fibonacci" -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22гиперболических+функций+Фибоначчи%22+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

762 results for "Hyperbolic functions" Lucas: https://www.google.com/search?q=%22гиперболических+функций%22+Люка+

182 results (%24 of total) for "Hyperbolic functions" Lucas -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22гиперболических+функций%22+Люка+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

3 results for "Hyperbolic function Lucas": https://www.google.com/search?q=%22гиперболических+функций+Люка%22

0 results (%0 of total) for "Hyperbolic function Lucas" -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22гиперболических+функций+Люка%22+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

889 results for "Hyperbolic functions Fibonacci and Lucas": https://www.google.com/search?q=%22гиперболических+функций+Фибоначчи+и+Люка%22

3 results (<%1 of total)for "Hyperbolic functions Fibonacci and Lucas" -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22гиперболических+функций+Фибоначчи+и+Люка%22+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

We can also perform this analysis in the Ukrainian language. Stakov, after all, is from the Ukraine. "Гиперболические функции Фибоначчи" -Стахов translates to "hyperbolic functions Fibonacci" -Stakhov without English word order, or "hyperbolic Fibonacci functions" -Stakhov with English word order in place.

1380 results for "Hyperbolic functions" Fibonacci: https://www.google.com/search?q=%22Гіперболічні+функції%22+Фібоначчі

1220 results (%88 of total) for "Hyperbolic functions" Fibonacci -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22Гіперболічні+функції%22+Фібоначчі+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

8 results for "Hyperbolic functions Fibonacci": https://www.google.com/search?q=%22Гіперболічні+функції+Фібоначчі%22+

2 results (%25 of total) for "Hyperbolic functions Fibonacci" -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22Гіперболічні+функції+Фібоначчі%22+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

Lucas translates to Лукас in Ukrainian:

5 results for "Hyperbolic functions" Lucas: https://www.google.com/search?q=%22Гіперболічні+функції%22+Лукас+

1 result (%20 of total) for "Hyperbolic functions" Lucas -Stakhov: https://www.google.com/search?q=%22Гіперболічні+функції%22+Лукас+-Стахов+-Стахова

No result for "Hyperbolic function Lucas" -Stakhov -site:trinitas.ru -site:goldenmuseum.com: https://www.google.com/search?q=%22Гіперболічні+функції+Лукас%22+-Стахов+-Стахова+-site%3Atrinitas.ru+-site%3Agoldenmuseum.com

The results for the full search term "Hyperbolic Fibonacci and Lucas Functions" in Ukrainian are overwhelmingly Stakhov's.

It would be fair to say that, considered separately, the Hyperbolic Fibonacci functions could shown to be notable if the value of articles not contributed to by Stakhov can be demonstrated. We cannot say the same for the Hyperbolic Lucas functions. I already previously voted to Move [this article] to Fibonacci Hyperbolic functions or Fibonacci functions. The above search analysis has solidified my position.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 23:22, 5 February 2013 (UTC)

Suggestion: I would like to add that Fibonacci matrices (1,900 results) and Golden matrices (1,660 results) are already studied in Western mainstream literature and they are related to the functions under question. Fibonacci matrices and Golden matrices may have been better articles for WIKIWIZDOM to start with, and neither article exists as of current. As it stands, both potential articles appear to be due for realization.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 01:39, 6 February 2013 (UTC)

• Delete A search on Google scholar for "Hyperbolic Fibonacci" yields no independent reliable references for hyperbolic Fibonacci functions. The study of phyllotaxis in botany is mainstream and the connection of certain kinds of phyllotaxy with Fibonacci and Lucas numbers and the golden ratio is well-established. But the connection of hyperbolic Fibonacci and Lucas functions with phyllotaxis, along with "metallic proportions" and a "hyperbolic world" comprise a fringe theory, in the sense of having few adherents among investigators of phyllotaxis. Finally, I'll note that what are called the hyperbolic Fibonacci sine and cosines in the article are already defined in the article Generalizations of Fibonacci numbers, where they don't merit a separate name and also don't have reliable sources referenced. The lack of notability of hyperbolic Fibonacci and Lucas functions suggests that this article should be deleted. Mark viking (talk) 04:19, 25 January 2013 (UTC)

Mark viking. The paper Generalizations of Fibonacci numbers has a link to the article on the Internet http://web.archive.org/web/20091027103713/http://geocities.com/hjsmithh/Fibonacc/FibWhat.html published online in 2004. That article dose in fact, described the hyperbolic Fibonacci sine and cosine. But the first time, a new class of hyperbolic functions was described in Stakhov and Tkachenko article, published as a preprint in 1988.  In 1993, these authors published a paper Stakhov AP Tkachenko IS. Fibonacci hyperbolic trigonometry. Proceedings of the Ukrainian Academy of Sciences, Vol. 208, № 7, 1993. , Pp. 9-14 (Russian). The Journal  Proceedings of the Ukrainian Academy of Sciences which is a very reliable source. Therefore, the priority in the introduction of the hyperbolic Fibonacci and Lucas functions belongs to Ukrainian mathematician Stakhov and Tkachenko (1993). So to prioritise the 2004 article http://web.archive.org/web/20091027103713/http://geocities.com/hjsmithh/Fibonacc/FibWhat.html over the Stakhov and Tkachenko’s article (1993) is incorrect and is a violation of scientific ethics. This article is based on Stakhov and Rozin’s article  On a new class of hyperbolic function. Chaos, Solitons & Fractals, 2004, 23 (2): 379-389. This article gives a detailed description of the theory of hyperbolic Fibonacci and Lucas functions and corresponding mathematical identities.WIKIWIZDOM (talk) 17:58, 25 January 2013 (UTC)

This article is based on Stakhov and Rozin’s article — that does indeed seem to be the cause of some of the problems other users are finding. This raises the issue of whether the article might be a copyvio [1]. Deltahedron (talk) 07:31, 27 January 2013 (UTC)

Deltahedron you can clearly see that it is not a copyvio, when I said this article is based on Stakhov and Rozin’s article I meant it is based (not copied) on their ideas, if other users have a problem they are free to add and improve the article as they see fit, is that not the whole idea of wikipedia.WIKIWIZDOM (talk) 08:24, 27 January 2013 (UTC)

Thank you for that assurance. Deltahedron (talk) 10:02, 27 January 2013 (UTC)

Notice on actual results on Google Scholar:

Results for гиперболическими функциями "Фибоначчи и Люка" -site:trinitas.ru -site:peacefromharmony.org -site:trinitas.pro on Google Scholar (18) (Google Translate)

"гиперболическими функциями 'Фибоначчи и Люка'" Translates to: "hyperbolic functions 'fibonacci and lucas'".

List of domains from participating Russian and Ukranian academic institutions, per Results for гиперболическими функциями "Фибоначчи и Люка" -site:trinitas.ru -site:peacefromharmony.org -site:trinitas.pro search on Google Scholar (Google Translate)

• http://www.uabs.edu.ua/ (Google Translate)
• http://www.sanu.ac.rs/ (Google Translate)
• http://www.nbuv.gov.ua/ (Google Translate)
• http://www.msu.ru/ (Google Translate)
• http://www.esrae.ru/ (Google Translate)
• http://tti.sfedu.ru/ (Google Translate) (redirected from http://www.tsure.ru/)

Sincerely, siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 02:17, 6 February 2013 (UTC)

• Comment There are two major issues with this article. The first is that the strictly mathematical content is rather unoriginal and has been rediscovered numerous times. The proper place for the material so-called hyperbolic Fibonacci and Lucas functions, which are trivial variants of the ordinary hyperbolic functions, would be at Generalizations of Fibonacci numbers. The second is that the application to Phyllotaxis appears to be a fringe theory, and the article is currently expounding this material as if it were established main-stream science. Deltahedron (talk) 20:19, 25 January 2013 (UTC)
• Keep but trim down - the general concepts appear to be real and notable, although some of the cruft may have to be removed. Bearian (talk) 23:44, 29 January 2013 (UTC)

Article update

I have added 3 new sections to the article in order to give more depth to key aspects of this topic.

1. Hyperbolic geometry of phyllotaxis
2. Generalised Cassini formula for the Fibonacci λ-numbers
3. Hilbert’s Fourth Problem

I hope this additional information will convince the critics in the notability and importance of this article as well as stop it being threatened by deletion.WIKIWIZDOM (talk) 23:20, 25 January 2013 (UTC)

• Delete. Old wine in new bottles. Much of this material is standard, but dressed up to look like a new discovery, and much of the rest is just crankery. —David Eppstein (talk) 06:59, 26 January 2013 (UTC)

David Eppstein can you elaborate on your comment, otherwise it sounds like a very vague opinionated statement. WIKIWIZDOM (talk) 08:17, 26 January 2013 (UTC)

Note: This debate has been included in the list of Science-related deletion discussions. • Gene93k (talk) 02:49, 27 January 2013 (UTC)

---

Relisted to generate a more thorough discussion so a clearer consensus may be reached.

Please add new comments below this notice. Thanks, MBisanz ^talk 00:16, 1 February 2013 (UTC)

---

I would like to re-post here my last message from the Talk page of this article, that has not had a reply from the "critics" since the 28 January 2013. Hope it helps in reaching the final decision regarding the importance of this article.

I admit that in Lucas's work (1878) and, probably, Barker's work (1968), the mathematical formulas similar to formulas for the hyperbolic Fibonacci and Lucas functions have been introduced. In these works, these mathematical formulas are not treated as a new class of hyperbolic functions. Therefore, we cannot find in these works the mathematical relations, that are in Table 1 (recursive properties) and especially in Table 2 (hyperbolic properties). In Wikipedia’s articles “Fibonacci numbers” and “Generalizations of Fibonacci numbers” we cannot find the identities that are listed in the Table 1 and 2 of the article. The statement, that these formulas are very simple and their proof can be made by students, cannot be used as an argument against the hyperbolic Fibonacci and Lucas functions. The Identities for hyperbolic Fibonacci and Lucas functions were first proved in the article by Stakhov AP Tkachenko IS. Fibonacci hyperbolic trigonometry. Proceedings of the UkrainianAcademy of Sciences, Vol. 208, № 7, 1993. , pp. 9-14 (Russian) published by the recommendation of the prominent Ukrainian mathematician academician Yuri Mitropolsky. Journal «Proceedings of the Ukrainian Academy of Sciences» which is sufficiently authoritative source and academician Yuri Mitroposky is a reputable mathematician; this demands a serious reaction to this article. In 1996, Prof. Alexey Stakhov made a speech «The Golden Section and Modern Harmony Mathematics» at the 7th International Conference "Fibonacci Numbers and Their Applications" (Austria, Graz, July 15-19, 1996). In this speech Stakhov outlined the foundations of the hyperbolic Fibonacci and Lucas functions. Stakhov’s speech attracted attention of Fibonacci-mathematicians and was selected for the publication in the book «Applications of Fibonacci Numbers" (see Stakhov AP. The Golden Section and Modern Harmony Mathematics. Applications of Fibonacci Numbers, Volume 7, 1998, pp. 393 - 399). Publication of this article is evidence that in 1996 the Fibonacci-mathematicians and American Fibonacci-Association recognised the introduction of a new class of hyperbolic Fibonacci and Lucas functions. Another proof of the recognition of this class of hyperbolic functions is the publication of the article by Trzaska, ZW On Fibonacci Hyperbolic Trigonometry and Modified Numerical Triangles. Fibonacci Quarterly. 34, 129-138, 1996. The editorial board of the most authoritative journal in this field «The Fibonacci Quarterly" did not have any doubt regarding the fact that this article contained original mathematical material, unknown to Fibonacci-mathematicians, despite the fact that before Trashka's article (1996) Edward Lucas’s article (1878) and Barker’s article (1968) published on a similar subject, but not the same one. It follows from this, that Edward Lucas’s work (1878) and Barker’s work (1968) are not repeated in Trashka’s article, which was published in “The Fibonacci Quarterly” by the decision of the editorial board. Finally, in 2009, «World Scientific" has published a book Stakhov AP The Mathematics of Harmony. From Euclid to Contemporary Mathematics and Computer Science. New Jersey,London, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai: World Scientific, 2009. - 748 p. http://www.worldscientific.com/worldscibooks/10.1142/6635. Chapter 5 is called Hyperbolic Fibonacci and Lucas functions. There are also 30 original articles of other authors, published in fairly reputable international journals between 2004 and 2013 as development of the article by Stakhov A, Rozin B. on a new class of hyperbolic functions. Chaos, Solitons & Fractals 2004, 23(2): 379–389. This fact is additional evidence that this subject is developing in the modern science and mathematics. It is important to note that another key theory in the evolution of this subject is "Bodnar’s geometry." This geometry shows that the geometry of phyllotaxis is a special kind of hyperbolic geometry, based on the hyperbolic Fibonacci and Lucas functions. This means that Nature uses the hyperbolic Fibonacci and Lucas functions. This is why, the hyperbolic Fibonacci and Lucas functions can be seen as a fundamental aspect of modern mathematics. The above stated has led me to the conclusion that the criticism of this article, by D.Lazard and Deltahedron and other editors, is not justified and prejudiced. This is not constructive criticism or help in collectively improving the article in the spirit of Wikipedia, it has only one intention - to remove the article by any means. This criticism is a contradiction to the undisputed recognition of the hyperbolic Fibonacci and Lucas functions by the American Fibonacci-Association and Fibonacci-mathematicians all over the world. I don't see any point in continuing this discussion in its current tone, to be honest this whole experience has left me with a rather negative feeling. By writing this article and spending my time, all I was trying to do is share worthwhile information for the benefit of the community, I accept this article required improvement like all others but instead of help I am confronted with cynicism and marginalisation. If you think that the deletion of this article and suppression of this information are beneficial and constructive, then I am not interested in continuing this discussion any longer... WIKIWIZDOM (talk) 18:48, 28 January 2013 (UTC)

WIKIWIZDOM (talk) 14:12, 2 February 2013 (UTC)

The comment criticism of this article, by D.Lazard and Deltahedron and other editors, is not justified and prejudiced is not a helpful argument in deciding whether or not to retain this article. Deltahedron (talk) 17:32, 2 February 2013 (UTC)

Deltahedron what would indeed be helpful is if you responded to my last statement and explain why you still think this topic is not worthy of a page in Wikipedia. Even though it has a long history and is accepted and well documented by reputable international mathematical publications (print and web) and mathematicians around the world. Besides the fact that this is not some mathematical oddity but a subject derived from the laws of Nature with a myriad of potential applications and is a basis for further discoveries linking to natural sciences. I simply don't understand how you fail to see this.WIKIWIZDOM (talk) 09:06, 5 February 2013 (UTC)

Answer: ....by paying more attention to his own select quotation rather than the main pattern of your argument. This fallacy is called "Wrenching from context".siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 19:54, 5 February 2013 (UTC)

• Delete as a combination of (a) material covered elsewhere and (b) WP:FRINGE material backed up by unreliable references. -- 202.124.73.54 (talk) 02:31, 5 February 2013 (UTC)

(Personal attack removed)

In reference to the personal attack: That IP address belongs to 3G Mobile devices. Not all IP addresses remain fixed to the same device. Notice how some the edits stem from years back. Please do not assume them as being from the same person.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 20:51, 5 February 2013 (UTC)

In any case, in response to the IP, there is also (c), material that applies neither to (a) nor (b). Whether that constitutes a small or large part of the article is irrelevant. The article can be trimmed and moved to a different article, and it does not have to be an old one.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 20:56, 5 February 2013 (UTC)

• Delete: (1) The mathematics is trivial, and most of it is covered elsewhere; (2) What is not covered elsewhere is not notable; (3) The connection to botany is a fringe theory. These have all been established above by other editors. Ozob (talk) 15:53, 5 February 2013 (UTC)
• Delete per MarkViking, and because although the general topic is notable this is not. Bearian (talk) 17:26, 5 February 2013 (UTC)

• Move to Fibonacci Hyperbolic functions or Fibonacci functions. Being a "consequence of known theories" hardly qualifies any mathematics as trivial. A significant portion of higher mathematics can be derived from a more elementary set of operations. Wolfram MathWorld has an entry on Fibonacci Hyperbolic functions (http://mathworld.wolfram.com/FibonacciHyperbolicFunctions.html). Hyperbolic Lucas functions are not quite notable at the moment however, so the article may need some pruning once it's moved. Regards, siNkarma86—Expert Sectioneer of Wikipedia
  ^{86 = 19+9+14 + karma = 19+9+14 + talk} 19:35, 5 February 2013 (UTC)

• Further comment:

“ Binet's formula expresses explicitly the Fibonacci and Lucas numbers through the golden ratio $\Phi ={\frac {1+{\sqrt {5}}}{2}}$, as a function of a integer variable n. Binet's formula may be written as follows:

$(1)\quad L_{n}={\begin{cases}\Phi ^{n}+\Phi ^{-n}&{\text{ for }}n=2k;\\\Phi ^{n}-\Phi ^{-n}&{\text{ for }}n=2k+1\end{cases}}$

$(2)\quad F_{n}={\begin{cases}{\frac {\Phi ^{n}+\Phi ^{-n}}{\sqrt {5}}}&{\text{ for }}n=2k+1;\\{\frac {\Phi ^{n}-\Phi ^{-n}}{\sqrt {5}}}&{\text{ for }}n=2k\end{cases}}$

”

“ Binet's formula, as written in (1) and (2), suggests to introduce the following hyperbolic functions of a real variable x.

• Hyperbolic Fibonacci sine

$(3)\quad {\rm {sFs}}(x)={\frac {\Phi ^{x}\,-\,\Phi ^{-x}}{\sqrt {5}}}={\frac {2}{\sqrt {5}}}\sinh(x\log \Phi )$

• Hyperbolic Fibonacci cosine

$(4)\quad {\rm {cFs}}(x)={\frac {\Phi ^{x}\,+\,\Phi ^{-x}}{\sqrt {5}}}={\frac {2}{\sqrt {5}}}\cosh(x\log \Phi )$

• Hyperbolic Lucas sine

$(5)\quad {\rm {sLs}}(x)=\Phi ^{x}-\Phi ^{-x}=2\,\sinh(x\log \Phi )$

• Hyperbolic Lucas cosine

$(6)\quad {\rm {cLs}}(x)=\Phi ^{x}+\Phi ^{-x}=2\,\cosh(x\log \Phi )$

”

While to the untrained eye this seems to be trivial, please keep in mind the significance of the Hyperbolic Fibonacci and Lucas functions goes beyond simply being a consequence of a recurrence relation. Can you say that all other recurrence relations have the form x^n ± x^(-n) divided by constant (or none at all)? An informed individual can clearly understand that, no, not every recurrence relation will have that form. Is this trivial? Of course it isn't.

“ The hyperbolic functions are:

• Hyperbolic sine:

$\sinh x={\frac {e^{x}-e^{-x}}{2}}$

• Hyperbolic cosine:

$\cosh x={\frac {e^{x}+e^{-x}}{2}}$

”

To think that this is trivial is to say that a majority of recurrence relations can have this mathematical structure. That is impossible.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 03:59, 6 February 2013 (UTC)

All very well, but is this a !vote for "keep" or "delete", or what? And why? Which Wikipedia:Arguments to make in deletion discussions are you making here? Deltahedron (talk) 07:30, 6 February 2013 (UTC)

Yes... just scroll up. You will find the answers to your questions.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 16:53, 6 February 2013 (UTC)

If I understand you correctly, you're saying that these recurrence relations lead to a cute formula, and therefore they are interesting. Count me unconvinced; nothing I see in the article is more than a simple application of well-known facts about linear recurrences. Ozob (talk) 11:12, 6 February 2013 (UTC)

Well, I didn't know a formula could be "cute". Perhaps you could elaborate as to why you think I am saying that?siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 16:53, 6 February 2013 (UTC)

Surprisingly, that the opponents of this article ignore a number of indisputable facts, which are evidence of recognition of the hyperbolic Fibonacci and Lucas functions:

1. The most authoritative journal in this field «The Fibonacci Quarterly" published in 1996 the article of Polish mathematician Trzaska, ZW On Fibonacci Hyperbolic Trigonometry and Modified Numerical Triangles. Fibonacci Quarterly. 34, 129-138, 1996. This fact is recognition of the hyperbolic Fibonacci and Lucas functions by American Fibonacci Association. It is difficult to assume that the editorial board of «The Fibonacci Quarterly" did not know Binet formulas and Lucas article (1878), which was published by the Fibonacci Association in 1969

2. In 1996, Prof. Alexey Stakhov made a speech «The Golden Section and Modern Harmony Mathematics» at the 7th International Conference "Fibonacci Numbers and Their Applications" (Austria, Graz, July 15-19, 1996). In this speech Stakhov outlined the foundations of the hyperbolic Fibonacci and Lucas functions. Stakhov's speech attracted attention of Fibonacci-mathematicians and was selected for the publication in the book «Applications of Fibonacci Numbers" (see Stakhov AP. The Golden Section and Modern Harmony Mathematics. Applications of Fibonacci Numbers, Volume 7, 1998, pp. 393 - 399). I would like to remind that the editors of the book Applications of Fibonacci Numbers, Volume 7, 1998 are the most famous Fibonacci-mathematicians GE Bergum, AN Philippou, AF Horadam. This is the additional evidence of the importance of the article “Hyperbolic Fibonacci and Lucas Functions”

3. From the moment of Binet and Lucas works (19 c.) Fibonacci-mathematicians did not notice Binet formulas and not interpret them as a new class of hyperbolic functions. This was done in 1993 by the Ukrainian mathematician Stakhov and Tkachenko (Stakhov AP Tkachenko IS. Fibonacci hyperbolic trigonometry. Proceedings of the Ukrainian Academy of Sciences, Vol. 208, № 7, 1993., Pp. 9-14). Since then, a new class of hyperbolic functions attracted for attention of mathematicians and is actively developing in modern science and mathematics.

In my opinion, the article “Hyperbolic Fibonacci and Lucas Functions” should be kept in Wikipedia. — Preceding unsigned comment added by KennKramer (talk • contribs) 23:43, 7 February 2013 (UTC)

— KennKramer (talk • contribs) has made few or no other edits outside this topic.

As a rule, all new math concepts initially met a resistance from the mathematical community. In this respect, the most famous example is the introduction of complex numbers . They appeared in mathematics in the 16th century. Many mathematicians considered that these new mathematical objects are very mysterious. Only after about 1800, a number of mathematicians, including Gauss, realized that the complex numbers have very simple geometric interpretation. They correspond to directed segments in the plane. Full recognition of the complex numbers as the most important mathematical objects can be attributed to the 30s of the 19th century.

Apparently, something similar happened with the introduction of the hyperbolic Fibonacci and Lucas functions. The researches of great mathematicians Binet and Luka in the 19th century prepared foundation for the introduction of a new class of hyperbolic functions based on the "golden mean." But if we will be honest, we must admit that Binet and Luka did not start the interpretation of their formulas (in particular, Binet’s formulas) as the hyperbolic functions. This was first done by the Ukrainian mathematician Stakhov and Tkachenko in 1993.

These functions got further development in the article of Polish mathematician Trashka, who translated Stakhov and Tkachenko‘s article into English and published this material in «The Fibonacci Quarterly" (1996). Stakhov and Rosin’s article (2004) introduced the so-called symmetrical hyperbolic Fibonacci and Lucas functions, which underlie the basis of the article "The hyperbolic Fibonacci and Lucas functions." This is a brief history of the hyperbolic Fibonacci and Lucas functions.

But why these functions met unexpected resistance from experts Wikipedia? In the Russian language there is proverb "Silenus hindsight." Wikipedia’s experts suddenly saw in these wonderful functions something, that they did not notice before, namely, their elementary quality (“exercises for students”) and their direct connection with Binet and Lucas researches. But Stakhov and Tkachenko do not deny the connection of a new class of hyperbolic functions with the Binet and Lucas works. Stakhov and Tkachenko’s merit consists in the fact that they are the PIONEERS in this field. They have introduced a concept of hyperbolic Fibonacci and Lucas functions. This concept has been recognized by modern mathematical community, including Fibonacci Association. I believe that the article "Hyperbolic Fibonacci and Lucas functions" should be kept at Wikipedia. — Preceding unsigned comment added by KennKramer (talk • contribs) 13:13, 8 February 2013‎

• KennKramer. In the history of mathematics, there is a strange tradition: the mathematicians (even very famous) do not always appreciate truly the mathematical achievements of their contemporaries. Examples - a huge amount. Let us remember one example, related to the topic of hyperbolic geometry. The famous Russian mathematician academician Ostrogradski gave in 19 c. sharply negative assessment of hyperbolic geometry by Lobachevski. Ostrogradskii called hyperbolic geometry "pseudoscientific theory" unworthy not only mathematician but even simple math teacher. The Russian Academy of Sciences does not like to remember this infamous case,. When we read some of the arguments of Wikipedia experts against the hyperbolic Fibonacci and Lucas functions ("exercises for students», “old wine in new bottles” and so on), there arises the impression that the history is repeating. I urge not make Ostrogradski mistake and to leave the article "Hyperbolic Fibonacci and Lucas functions" in Wikipedia. — Preceding unsigned comment added by 70.61.223.22 (talk) 17:28, 8 February 2013 (UTC)

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