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Looking for a proof! (corrected)

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Looking for a proof! (corrected)

Dear PMs
The link:

contains a sub-link about looking for a proof !!:

Can you help on this? (as a math challenge or a simple problem)?
PS. those links can help:

It is probable that the subscript 'n + 1' on the RHS be 'n + j'.

Sorry, or 'n + j + 1'.

Thank you.
can you correct there?

Hey Zhangaini,
> can you correct there?
You are looking for a proof. How may one correct a mathematical proposition if one pretends to prove an unknown proposition? Logic says us that your above question has no answer.

there is no error.
The result is true, but no proof has been provided,
The request of proof stands.
Best regards!

Now your question makes sense. Let's recall the first relation.

B_{n+1}(x)B_{n+j}(x} - B_{n+j+1}(x)B_n(x) = (3x^2+4)D_{n+1}(x,1/4).

I suppose 'j' ranges over nonnegative integers. Let's take, e.g. the first one j=0. Thus for j=0,

B_{n+1}(x)B_n(x) - B_{n+1}(x)B_n(x) = (3x^2+4)D_{n+1}(x,1/4),


(3x^2+4)D_{n+1}(x,1/4) = 0.

But this equation is absurd as neither (3x^2+4) nor D_{n+1}(x,1/4) vanish identically. Do you want discard j=0? Not problem!

For j=1,

B_{n+1}(x)B_{n+1}(x) - B_{n+2}(x)B_n(x) = (3x^2+4)D_{n+1}(x,1/4). (1)

For j=2,

B_{n+1}(x)B_{n+2}(x) - B_{n+3}(x)B_n(x) = (3x^2+4)D_{n+1}(x,1/4). (2)

Next, if you subtract (1) from (2), the RHS is zero because (3x^2+4)D_{n+1}(x,1/4) is invariant with respect to 'j'. So that, leaving the positive terms on LHS and passing the negative terms on the zero RHS, you will get, in each side product of different Boubaker polynomials each other. However, all Boubaker Polynomials are different, arriving so to a contradiction. Moreover, one can choose any pair of nonnegative integers which makes the contradiction more evident.
On the contrary, the second equation

B_n(x) = D_n(2x,1/4) + 4D_{n-1}(2x,1/4),

at least is consistent. Note that this equation is the reciprocal recursion of the first equation, which is a strong proof that the first equation is incorrect.

Hi Zhangaini,
There is a mistake in the LHS of the first equation, since assigning different values to the subscript 'j', the RHS doesn't change which is absurd. Please, check it out.

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