Happy New Year;

what is the value of n? 

The number n is a natural number not less than 2.

Well, i checked for n =2, it's correct! I guess we can use induction on n, l and m! 

 Pour n=3 le resultat est correct ,reste  à prouver par recurrence  pour n superieur à 2 

Dear Messahel Abdelkader

Could you please write in English? Thank a lot.

Mr. Abdelkader proved it for n = 3, the issue is a 3 loops induction over l, m and n

What is a 3 loops induction?

index: JD_RG_201612291210

General remarks:

 - Feng Qi's equation  EQN(m,n)  is independent of  $\ell$

 - the question is the validity of the EQN(m,n)  for all pairs  of naturals  n  and  m  such that  n \ge 2 m \ge 2.

Consequently, the induction can be applied at most to two prameters. For instance, with only one "loop" or with two "loops" as follows   (  =>  means "implies"  )

PROPOSED SCHEME1 OF THE PROOF

For every fixed  m prove the facts

EQN(m,2m) is true

EQN(m,n) implies  EQN(m,n+1)   for  every  n \ge 2m   (this step is probaly what Jamal Salah has meant as the induction loop)

PROPOSED SCHEME2 OF THE PROOF

Step1: prove the facts:

EQN(1,2) is true

EQN(1,n) => EQN(1,n+1) for every n \ge 2

Step2: prove the fact that

{[ EQN(m,2m) is true] AND [ EQN(m,n) => EQN(m,n+1) for every n \ge 2m] }

=>

{[ EQN(m+1,2m+2) is true] AND [ EQN(m+1,n) => EQN(m+1,n+1) for every n \ge 2m+2] }

This is a little bit messy, though, not the unique way.

Unfortunately, no concrete facts are known to me (till now, at least:). Hopefully the remarks may help.

Best regards

Dear Feng,

Happy new year to you and all RG members.

Messahel Abdelkader wrote For n=3 is the result valid, we have to prove it by induction for n>2. 

All contributors are proposing induction. Ok, but only induction in n, starting with n=2m. This could work without specifying m.

Nevertheless, I continue to hope that another proof based on a +/- known identity for combinatorial numbers will be found.

The following identity is necessary for Feng Qi's equation:

  (*)     $\sum_{\ell=0}^{m-1} \, \left(-{1\over 4}\right)^\ell \,\left[ {2m \choose 2} - (2m-1 -\ell) \right]  \frac{(2m - \ell -2)! }{l! \, ( 2m - 2 \ell -1)! }=0 , \; \mbox{ for } \; m=1,2,3, \dots $

For details and comments see the attachment, please.

Joachim, can you please include the sign´=´ into your (*)? 

Brackets in '[2(...)]!' would also help. Thanks :)

Viera, you are right, of course, Thanks, and Happy New Year'[17]!

PS1. The brackets are added according to your remarks.

PS.2.  The Feng identity is obviously true for  m=1, n\ge 2, since then  the unique term equals zero. Moreover, I have positively verified  the identity  for  m=2  and  m=3, with arbitrary   n\ge 2m, since the rational functions of variable  n  were very simple:)

@ Feng Qi

Dear sir, would you please inform our RG community, how you have obtained the identity, whether it is known from some sources (possibly  without proof), and why you are interested in it.

In my case, your explanation is needed to get motivation for further advices. For instance I have got a confirmation of the necessary  equalities (*) from my answer (no. 10 on this post) via generating functions. Are you  interested in such partial results?

Sincerely yours

I have reformulated a little. I do not have a proof but I hope this can be of some help.

See attachment. Best wishes.

 Hi, here is a proof, first obtained by the computer (using an implementation of Gosper's algorithm in Maple), and written here independently.

Regards,

@ Frédéric Chyzak 

Well done!  Congratulations!

With best regards,

Joachim Domsta

I am reading Chyzak's answer carefully. I will be back soon after I understand Chyzak's answer. Thank a lot.

Dear All

The answer to the problem is at

http://dx.doi.org/10.20944/preprints201703.0209.v1

or

http://www.preprints.org/manuscript/201703.0209/v1

Feng

@ Feng Qi

Dear sir,

can you please release an evaluation of the proof by Frédéric Chyzak, which seems to be correct. In  your paper the same  equality is proved as Theorem 5.3 (in a different way; but all proofs of the same theorem are formally equivalent!).

However,  there is no remark about the nice advice by Frederic, which should be acknowlegded if it helped, or if Frederic was the first author of such a proof. Of course, if  you  have discovered a substantial gap in Frederic's proof, our RG-community would be highly interested about this, please.

Sincerely yours, Joachim Domsta

Dear Joachim

I remebered Chyzak's proof for this question. I scanned Chyzak's proof, but not checked up it carefully. In theory, all proofs of the same theorem are formally equivalent to each other. But in practice, the gap between different proofs is generally very large. So sometimes we need various and different proofs for the same theorem in mathematics. Chyzak's proof is too complicated in my eyes, so I scanned but did not check up it. You can see that the proof of my Theorem 5.3 is indeed a consequence of combining Theorem 2.1, Theorem 3.1, and Theorem 5.1. We can not say my proof is equivalent to Chyzak's. In other words, I did not influence by Chyzak's proof when I wrote this manuscript. Moreover, in my opinion, Chyzak's proof is not suitable to be put in my manuscript, it has a different style or taste from my manuscript. Therefore, I did not mention Chyzak's proof in my manuscript. Hence, I did not acknowledge Chyzak and any other persons (I ever asked some other questions eslewhere and many colleagues replied me). If someone insists that I should acknowledge Chyzak and mention his proof in my manuscript, if Chyzak would like me or allow me to do so, I would do by adding a remark or an acknowledgement in a revised version.

Carefully reading my manuscript, one will find that Theorem 5.3 (Chyzak's proof) is not the main aim of my ideas in the manuscript. The manuscript should be regarded as a whole. As a whole, I do not think that Chyzak's proof should be put into my manuscript as a main or alternative proof of Theorem 5.3.

Carefully reading my manuscript, one will understand that Chyzak's proof is not what I needed in my manuscript.

Due to the above reasons, I did not mention Chyzak and his proof for Theorem 5.3.

I repeat here that, if someone insists that I should acknowledge Chyzak and mention his proof in my manuscript, if Chyzak would like me or allow me to do so, I would do by adding a remark or an acknowledgement in a revised version. However, currently I do not think it is a better choice to put Chyzak's proof as an alternative one of Theorem 5.3. The better choice is that, after extending, Chyzak publishes his proof independently somewhere.

Anyway, thank Joachim for your kind and valuable suggestions.

Best regards

Feng

You may have eventually found another proof of the identity, one that suits you better. That is fine and normal. I also think, however,  that you should mention or acknowledge in some form Chyzak's proof. That is my opinion.

Best regards.

Stefano

Before I occationally invited some colleagues to be as coauthors of several papers for their kind suggestions, careful corrections, valuable comments, or essential contributions to my manuscripts, but I were declined several times. Frankly speaking, I feel it is more difficult to handle such a problem than mathematics.

Dear  Frédéric Chyzak, how about your opinion?

Dear Feng,

I understand your point of view, BUT if someone is asking publicly for the proof,  then  you are getting involved in some cooperation and duty with respect to anyone who answered you. By not even checking the supplied ready to use proof, not taking into account the lot of time spent by Frederic to

A. discover the proof (with computer  programming)

B. check the proof

C. formulate the proof in a publicly acceptable form,

you have obtained at least the following advantage:  The theorem is OK.

Not reading the details of the proof you are also saying:  It doesn't matter what you have done, I don't need you anymore, looks like saying "you are the looser, your work was senseless".

NO, this is anacceptable. It is not necessary to get involved deeper than within this pages, however more interest exposed to the job made FOR YOU is obligatory. So, at least check the proof by Chyzak. Do you think he does not wait for your evaluation of his work, and even for some comparison to your proof? A remark devoted to the published proof within RG is also important for the helper. Who is talking about co-athoring? Also, you don't have to publish in journals Chyzak's proof. Just mention that it helped you in getting confidence of correctess of your proof, DESPITE THE CORRECTNESS OR INCORRECTNESS OF HYZAK'S PROOF!!!!!!!

Regards, Joachim

If I used or mentioned Chyzak's proof but did not mention him, didn't acknowledge him, didn't invite him to be a coauthor, or didn't refer to a reference, I can be sure that I made a mistake.

Dear RGaters,

RG is NOT a rescue service. Thus asking a question we cannot leave the community of involved followers just because our health  got improved in the meantime. It is even not enough to check out. Some attention is required to ANYONE who answered, for her/his contribution (perhaps excluding cases of abuse, of acting with bad goal, of too impolite claims etc).

Such a problem becomes a real problem! I do not know how to deal with it. Why does one must understand what he/she cannot understand? What I wrote above means that I did always cautiously and carefully before. I would like to hear opinions from all colleagues in the world. If Frédéric Chyzak allows or requires, I will do according to any suitable requirements by him. In my opinion, this problem should be dealt with simply and reasonably.

Dear Professors Joachim Domsta and Stefano Capparelli

I really do not know how to copy with Chyzak's proof. Under such a situation, I would like to adopt your suggestions. For this, I will contact Frédéric Chyzak by e-mail immediately to one among the following ways

(1) Combine Chyzak's proof into my manuscript as a second proof of Theorem 5.3 and invite him to be a coauthor of this revised manuscript

(2) Add a remark to mention that Chyzak supplied on 15 January 2017 an alternative proof of Theorem 5.3, but do not recite his proof in details

(3) Add an "Acknowledgements" to acknowldge that Chyzak supplied on 15 January 2017 an alternative proof of Theorem 5.3, but do not recite his proof in details

(4) Other way Chyzak would like

One more question: should I acknowledge everybody who ever wrote here something relating to the identity? For example, Jamal Salah, Messahel Abdelkader, Viera Čerňanová, and including you two? I really need your advices about it.

Best regards

Feng Qi (F. Qi)

Dear friends, thanks a lot for your interventions. 

Namely you, Joachim, you are here doing important service to preserve fair dealing between RG members. 

Let me please join the discussion. 

As a recent co-author of professor Feng Qi, I would like to add some words. As I know him, he is an extremely correct person. I remember a situation, where he asked me in private communication for an advice: he aimed to propose co-authorship to 5 or 6 contributors replying to one of his question in RG. Feng Qi asked me: "What can I do, it some of them does not accept?"  I understood that this would be very embarrassing for him.

Notice, Feng wrote above: "I were declined several times. Frankly speaking, I feel it is more difficult to handle such a problem than mathematics." 

Here enters not only the fine personality of Feng Qi, but also a great difference in our cultures, that of the East and the West.

For instance, if we, people from the West, knew, how shocking are 

• ???  
• or  ! 
• or even !!!

in our messages for them... we would use much more carefully these punctuation signs. Except in factorials :)

Dear Feng, surely do not mention my person in this paper. 

I propose you to contact Frédéric Chyzak by e-mail.

I would vote for Joachim Domsta as well, because his participation in December seemed to me valuable.

You can eventually do private communication with other contributors.

Good luck and best wishes, 

Viera

Dear All

I am preparing an "Acknowledgements" for the revised version of the following preprint. See the picture or the PDF file uploaded. Sorry for my neglect in the first version. Hope to hear your advices soon.

Feng Qi and Bai-Ni Guo, Identities of the Chebyshev polynomials, the inverse of a triangular matrix, and identities of the Catalan numbers, Preprints 2017, 22017030209, 21 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0209.v1 or at http://www.preprints.org/manuscript/201703.0209/v1.

Feng

Dear All,

Thanks for every advice leading to better communication between people. I understand pretty well the difference between Cultures, if there are any. My point is, however, that some suggestions made by me are aimed to onepoint, which should be common: If someone starts a problem and the responder asks a question back, then the author of the problem should continue the merital discussion. As I said before, it is not a problem of co-authorship or even the RG-points, since advice's aim is usually to help in getting closer to the right answer. Therefore my suggestion was just to say suitable "thanks" for the proof. That's all. If this is not expressed, then the question appears: what all this is for? People are trying to do the best in both, the content and the form, and later on get a message:

The answer to the problem is at . . .

without even checking if the supplied follower's proof is correct. This is what I criticize as unacceptable. Further detail is the following: Is the Frederic proof really so difficult to be understood? 

Another aspect from Viera's answer about proposal of co-authorship: "I understood that this would be very embarrassing for him."   Obviously this applies to all participants asked for being co-athor, as well.

Dear Feng, please accept the critic as follows: people helping by giving answer in public, especially if it is complete, expect at least that heir answer is studied as much as possible; if somethig is not understood, then an  answer is expected with indicating  places requiring further explanation. Without this there is no discussion. Even worth, people may feel exploited in the bad meaning of the word. It is no excuse that you didn't use or presented the other proof. You have asked a question, you have got a valuable answer,  this makes Frederic first who proved your conjecture, and no trace remains in the literature. Of course, Frederic still has right to  publish the proof of your conjecture. But according to the  citations custom, you will be the first. Not mentioning about the facts  in your publication, taking into account the entity of the circumstances you are obtaining the first place without deserving it. This is my main objection.

PS. Please do not mention my contribution in your paper, since you didn't notice it. At least, you didn't answer to my proposals, if they are correct, or wrong; were they helpful or not;  you even didn't answer questions asked in my answer no. 13.

Sincerely, Joachim

This  formula is true,I verified it  for 2

Dear Messahel Abdelkader

Would you like to be as a coauthor of the manuscript by adding your verification for 2

Dear Joachim Domsta

I feel difficult to understand you and your long english sentence. For example, where is "in my answer no. 13"? How do you count on the number 13?

I respect your declining to be acknowledged in the manuscript, I will remove your name from the list that will acknowledged.

Till now someone accepted my anknowledgement, someone declined, someone is considering, and someone do not reply me. You are an elder and have rich experiences, could you please tell me how I should do for those colleagues without reply?

As for your answer to the mathematical problem, I noticed your answer, but I have no deep reading, because I feel a little difficult to understand your english sentence and mathematical language. In my opinion,I can not put what I can not or do not understand into my manuscript. So I did not consider to put your answer into my manuscript and even to respond to your answer (I do not remember clearly). I like simple mathematics and language.

Right now I have to teach 130 undergraduates, they are waiting for me to teach their ComplexTheory. See you later.

Sincerely

Feng

Dear Joachim

How are you? Just now I finished my two-hour class.

Because you wrote too long and your english sentences are too long, I have to understand one sentence by sentence.

I cannot understand this sentence "But according to the citations custom, you will be the first. Not mentioning about the facts in your publication, taking into account the entity of the circumstances you are obtaining the first place without deserving it."

What is the meaning of the sentence "But according to the citations custom, you will be the first." I will be the first? I do not think so, never think so. I posed a mathematical problem publicly. If someone solves the problem, it has nothing to do with me, provided that the solver mentions the origin of the problem. If not mentoning me and the origin of the problem, I do not mind anything.

What is the meaning of the sentence "you are obtaining the first place without deserving it" Do you mean I do not deserve to be "the first"? In my opinion, you used a wrong word "deserve", because Chyzak's proof belongs to Chyzak, not to me, so I have no right and no desire to get "the first".

By my understanding, I can not put what I can not or do not understand into my manuscript. I can not or do not wrote material I think unuseful into my manuscript. Except Chyzak's proof, perhaps your answer is also correct, but I can not understand if it is correct, so I can not put your answer into my manuscript, so I did not know if I should acknowledge you in my manuscript. There are so many persons spoke here, some ideas are correct but unuseful, some ideas are correct but can not be carried out, some ideas are wrong, some ideas are correct but I can not understood, and the like. Therefore, I must judge and choose those which are possibly useful to my work. For this, I must scan them one by one quickly and choose one or more to consider and to understand. If finding the useful one, I think I would not consider others. If I cannot find any useful one, I would consider to try by my own ideas. My ability and knowledge is limited, it is very possible that I can not or do not choose the best one. Under such circumstances (this word was used by you, but I did not understand accurately till now), should I be criticized as you did?

On the other side, the ResearchGate is an academic forum. Generally there are many interesting mathematicians or scientists talking and discussing a "Question". Honestly speaking, I have no time and ability to respond every answer. Except me, everybody can respond others. What is the meaning of the word "forum"?

In my opinion, we should make it simple, not make it more and more complicated and complex, because almost everybody is busy with his/her own duties. I posed this "Question" to ask for help solving a mathematical problem. While you did, I did; while I did, you did. Finally you said you solved (although I can not or did not understand), I solved too. I completed a manuscript without adopting any of your solutions. I announced the mnuscript (preprint) on my own initiative here. You thought I should at least acknowledge some solvers, even everyone, but I did not think so when I composed the manuscript (because I did not adopt any of your solutions). You suggested me to at least acknowledge some solver, even every talker here. I accepted your kind suggestions. Right now I have been inviting somebody, including you (but you declined), who I think they deserve (this is your word) it, to be as coauthors, or asking for permits of others to be acknowledged in my manuscript.

Do you remember that, because I adopted in the manuscript below some material you provided on the ResearchGate, I invited you to be as a coauthor of the following manuscript?. You declined to be as a coauthor and permited me to acknowledge you in it. I did as you would like.

Feng Qi and Bai-Ni Guo, Levy--Khintchine representation of Toader--Qi mean, Preprints 2017, 2017030119, 10 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0119.v1.

This is the whole story. This story is so simple. I wish it becomes simple. I wish this story ends here, stops here, and finishes here.

Best regards

Feng

Dear Feng,

my contribution was negligible. please do not mention my person in your paper Identities of the Chebyshev polynomials, the inverse of a triangular matrix, and identities of the Catalan numbers. 

Thanks, Viera

Dear Feng,

Thank you for the questions, and explanations. Let me answer as briefly as possible:

The current discussion is about this current question, only.

My first objection is that you didn't notice, even within this sequence of answers, that Chyzak attained a roght proof of the equation post by you; you could at list say "thank you" and stating that you are not using it in your further research, but, that you appreciate his contribution.

The second objection is that publishing your own proof (no doubt about this) before Chyzak, according to the usual custom of counting achievements you will be the first as the author of the proof, not Chyzak. This is because RG announcements do not count as publication. And this effect, that you would be the first who published the proof is not justified. Therefore I have suggeted also to mention in your paper, that there is another proof earlier presented to you by Frederic Chyzak.

No 13  means the third answer on page  nr 2 in this series of answers to your question. Lack of answer to it and to other answers is just breaking a good custom. By the way, I have mentioned this with respect to my answers only because you wanted to aknowledge my contribution, which - let me stress again - was not justified due to your decision of not studying  it. Obviously, I respect this  entirely. To close this aspect: I decline again to be mentioned in the current paper with Theorem 3.1. This is in no relation to the paper on Levy- Khintchin representation.

I never sggested (or I was wrongly understood) to acknoledge everyone who discussed the question. Chyzak's answer is a particular exception, since this is exact and complete answer with a very tricky proof (AND STILL IT IS A PARTICULAR EXCEPTION)

What to do with people who do not answer to your proposals? I cannot help you.

Sincerely, Joachim

PS. I also hope that this closes the non-math problem. JoD

Yes, i would gladly accept

Dear all,

I thank Prof. Feng Qi for bringing his problem to our attention, as well as all those who have spoken so as to maintain good citation practices between us in the research community.

Having read all your comments as well as Qi and Guo's preprint, I think it would be fair to me if the authors just made a remark that I suggested an alternative proof based on Gosper's algorithm.

To speak a bit of my motivation in answering Prof. Qi's original question, I immediately realized that it could be an application of the method developed by Gosper in his 1978 article "Decision procedure for indefinite hypergeometric summation". This article can be obtained there (also as a pdf file):

http://www.pnas.org/content/75/1/40?related-urls=yes&legid=pnas;75/1/40&cited-by=yes&legid=pnas;75/1/40

By the way, equation (3.4) in the preprint, which provides a sum of the Chebyshev polynomials of the second kind equal to n (2x)2n, can be obtained by an extension of Gosper's algorithm that was developed in the 1990s.

Best regards,

Frédéric Chyzak

There is no need to acknowledge my negligible contribution.

Best wishes,

Stefano

Dear Messahel Abdelkader and Frédéric Chyzak

To Messahel Abdelkader: I noticed the sentence "Yes, i would gladly accept". Do you mean you accept my invitation to be as a coauthor of the revised manuscript? If yes, write me to [email protected]. If no, I have the following suggestions and comments.

To Messahel Abdelkader and Frédéric Chyzak:

(1) Till now possibly there are three (complete, independet, different, unequivalent) proofs for the identity I posed (currently I do not know what Abdelkader's proof is): Chyzak's proof by Gosper's algorithm, Abdelkader's proof, and mine. As a proposer of the (not so important) identity and one of its solvers, I think that we have no need to mind which proof is earlier or later, because I believe these three proofs should be independent (but, in theory, euqivalent) and the time difference is not so large in "mathematical time system". This way has always been adopting by the American Mathematical Monthly to cope with many solutions (answers) and their solvers of problems issued in Monthly each issue. Since you declined to be as a coauthor of the revised version, I cannot put your two proofs (including Chyzak's potential proof for (3.4) by an extension of Gosper's algorithm) into my revised manuscript. In my opinion, putting your proofs, including Chyzak's possible proof for (3.4) in the preprint we are discussing, into the revised version would change the structure and style of the manuscript. This means that you can freely handle your proofs. My first suggestion is that you publish your proofs, including Chyzak's proof for (3.4), separately, independently, and respectively.

(2) The second suggestion is that you can consider to combine your two proofs, including Chyzak's proof for (3.4), into the same manuscript and publish it somewhere.

(3) Perhaps other proofs supplied by other colleagues spoke in this "Question" are also correct, or can be carried out, or can be realized.

(4) It is very possible that the identity and its proof(s) have existed somewhere.

That's all.

Thank you all and other colleagues for your contributions to the proofs of the identity.

I am very pleased for proposing a question motivating so many talks and discussions. I believe that there will be more and further developments caused by this "Question".

By the way, along with time flying, I feel stronger and stronger that, sometimes writing down a correct conclusion is more difficult than proving it (for example, the inverse matrix in the preprint), sometimes writing down a correct conclusion is easier than proving it (for example, Goldbach‘s conjecture), sometimes both are difficult (for example, the Gauss--Bonnet theorem to n-dimensional Riemannian manifolds), and sometimes both are easy.

Best regards

Feng Qi (F. Qi)

https://qifeng618.wordpress.com

Feng Qi, Qing Zou, and Bai-Ni Guo, Identities of the Chebyshev polynomials, the inverse of a triangular matrix, and identities of the Catalan numbers, ResearchGate Working Paper (2017), available online at http://dx.doi.org/10.13140/RG.2.2.27344.71684.

Feng Qi, Qing Zou, and Bai-Ni Guo, Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers, Preprints 2017, 2017030209, 25 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0209.v2 or at http://www.preprints.org/manuscript/201703.0209/v2.

A reorganized presentation in Chinese for this preprint has been posted as

祁锋，与第二类Chebyshev多项式和Catalan数相关的一个矩阵之逆和几个恒等式， ResearchGate Presentation (2017), available online at https://doi.org/10.13140/RG.2.2.22832.46081/1

The following formally papers are related to this question:

[1] Feng Qi, Qing Zou, and Bai-Ni Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Applicable Analysis and Discrete Mathematics 13 (2019), no. 2, 518--541; available online at https://doi.org/10.2298/AADM190118018Q

[2] 祁锋，一个三角矩阵之逆与Catalan数恒等式, 湖南理工学院学报(自然科学版)，2020年第33卷第2期，第1页至第11页转第22页；available online at https://doi.org/10.16740/j.cnki.cn43-1421/n.2020.02.001