How to prove or where to find a proof of the lower Hessenberg determinant showed by two pictures uploaded here?
Dear Feng Qi ,
I have implemented the algorithm, and the answer is indeed to obey the mentioned formula.
The answer to your question is not direct.
I tried the well- known techniques, the mathematical induction and Shurs complement determinant formula over the block entries.
Where Hlxl =[ H(l-1)x(l-1) , X
[ Y^T nz ]
where the entries of X and Y are known vectors.
But before we are doing the details, I have the following question:
It is not clear how you proof this result, is it your work? If so, logically you know the formula proof, otherwise, what is the source of your formula?
Best regards
Dear Professor Feng Qi
I do not know if this is known, But I managed to produce a proof of this determinant formula
Please see the attached file
Omran Kouba Issam Kaddoura
Dear Professors Issam Kaddoura and Omran Kouba
Thank you very much for your interesting in my question and for your nice answers.
Frankly speaking, I have also obtained an indirect proof for the formula (3.3) when I announced this formula and asked this question here. Motivated by another paper, I derived and wrote down the formula (3.3). On 5 June 2019, I submitted the formula (3.3) and my indirect proof to a SCIE mathematical journal for possible publication, but I have no confidence in its acceptance.
Judging from your answers and my derivation, I think that the formula (3.3) and its proofs should be not trivial.
I have a suggestion or an invitation to you: in my opinion, it is better to combine your proof and algorithm into my manuscript and jointly publish the formula (3.3) and its proofs and algorithm somewhere. If you agree to my suggestion or would like to accept my invitation, please send your files (TeX or text) to my e-mail box ([email protected]).
Anyway, I appreciate your direct proof and algorithm of the formula (3.3) and your interests to my question.
Best regards
Feng Qi (F. Qi)
https://qifeng618.wordpress.com
Issam Kaddoura
Two e-mails I sent to your [email protected] several minutes ago were bounced back.
Issam Kaddoura
I resent two e-mails to your email box [email protected]. Hope to hear from you through e-mail soon.
Sorry for my typo on your e-mail address.
Dear Feng Qi ,
I think this has a new form and (alpha) is a positive integer.
Best regards
Dear Professor Issam Kaddoura
As it is clear from my answer above (the proof of the corollary is included), (alpha) may be any complex number. In fact if this holds for positive integers then it holds for all complex numbers because both sides are polynomials in (alpha).
Dear Professor Omran Kouba ,
Yes, I agree, it is compatible with the notations included in your previous answer.
Best regards
Dear's Feng Qi and Omran Kouba , It was a great opportunity to work with you. Indeed both of you show excellent cooperation and professionalism in this domain of research. Hope to cooperate again in the future. Best regards
Could you please send me the paper, so that I can see, what you wrote ?
Dear Prof.Qi, One idea is given in the joint paper Tomovski,Pogany,Srivastava, J. Franklin Institute, 2014, see Laplace type integral expressions for a certain three-parameter family of generalized Mittag-Leffler functions with application
- 10.1016/j.jfranklin.2014.09.007 Best wishes
Zivorad Tomovski
Dear Professor Tomovski
I scanned your mentioned paper, but I did not find any idea related with this question discussed here. Could you please supply more details including pages range, line numbers, and the like? Thank you very much.
Best regards
Feng Qi (F. Qi)
https://qifeng618.wordpress.com
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