In the preprint
W.-H. Li and F. Qi, A further generalization of the Catalan numbers and its explicit formula and integral representation, Authorea Preprints (2020), available online at https://doi.org/10.22541/au.159844115.58373405
I concluded two integral formulas indicated in the picture.
(1) Do you know the existence of these two integral formulas? Please give concrete and explicit references containing these two integral formulas.
(2) Can you find direct and elementary proofs for these two integral formulas?
Dear Feng Qi
I am attaching a proof of the first formula.
I see setting p equal to one half in integral formula (5.1), then rearranging, yields integral transform (5.2)
Kindly check https://www.toppr.com/guides/maths/integrals/integral-of-some-particular-functions/
Finding the Integral of a Product of Two Functions
Declare a variable as follows and substitute it into the integral: Let u = sin x. ...
Differentiate the function u = sin x. This gives you the differential du = cos x dx.
Substitute du for cos x dx in the integral:
Now you have an expression that you can integrate:
Substitute sin x for u:..
https://www.dummies.com/education/math/calculus/finding-the-integral-of-a-product-of-two-functions/
Misleading answers disturb serious followers!!
One needs to read the question carefully, and then post an answer, if any.
To post silly answers seeking the highest RG score, make one nervous!
Dear Feng Qi
It seems a deep interesting formula. It is very close to the elliptic integrals of the
(first, second, third!) type.
I think you have the proof otherwise, it is impossible to guess a random answer
for such integral. Can you mention the type of the posted integral?
Regards
@ Feng Qi
I would suggest using complex integration methods. You would essentially sum over all residues. The transformations for the product and sums, including binomial coefficients, is a lot of work. Scour through integral tables and also relationships between binomial coefficient definitions to find manipulations sought.
Dear Professor Omran Kouba
Technically your proof is different from mine in the mentioned preprint. However, your proof and mine in the preprint are the same: they all use Faa di Bruno formula (with different forms) and Cauchy’s integral formula. Even so, I think your proof is very nice!
Dear Feng Qi
yes you are right there is a lot of similarities, I must learn to read the details of the question first, I saw the article you mentioned after having written my answer, I think the use of Faa di Bruno is mandatory here. Thanks
Dear Professor Feng Qi
Thank you, I will be honored, would you like me to send the latex version by email?
Dear Feng Qi ,
I sent you the the latex code by email to you email on hotmail.com.
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