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Revision History for A174227

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

Expansion of -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x).
(history; published version)

#19 by Peter Luschny at Fri Nov 04 17:09:22 EDT 2022

STATUS

proposed

approved

#18 by Georg Fischer at Fri Nov 04 15:47:11 EDT 2022

STATUS

editing

proposed

Discussion

Fri Nov 04

16:47

Peter Luschny: I don't understand. You says (in Maple) 
ogf:= -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x).
This is wrong. See the output at 10:05.
The corrected definition/name says ogf := (1-10x+sqrt((1-10x)(1-14x))/(2x).

16:48

Peter Luschny: (1) Only one of the two can be correct. (2) How can you arrive at the correct 
result with an incorrect definition? There is something deeply wrong here.

16:57

Peter Luschny: Well, now I see that you changed the definition in edit 15:47. So (1) is OK now, but still I remain suspicious because with the wrong definition  your construction obviously worked.

16:57

Peter Luschny: Which I can't verify because I don't have Maple 2022.

17:00

Georg Fischer: Be assured that it also works with the new definition.

17:05

Peter Luschny: If I understand you correctly, you are saying that your method proves recursion. And for that all the effort. Of course you don't see this in the text, and you know it only if you are owner of Maple 2022.

17:09

Peter Luschny: Uff, in 16:47 I copied the wrong line. Sorry. Which does not change the argumentation.

#17 by Georg Fischer at Fri Nov 04 15:42:14 EDT 2022

MAPLE

ogf:= (1-(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x): initreq:= [1, 1, 12, 145, 1764, 21602]:FindRE(ogf, x, u(n));

init:= [1, 1, 12, 145]: iseq:= seq(u(i-1)=init[i], i=1..nops(init)): req:= FindRE(ogf, x, u(n));

EXTENSIONS

Definiton corrected by Peter Luschny, Nov 05 2022

Discussion

Fri Nov 04

15:47

Georg Fischer: The ogf was syntactically incorrect and did not work, but it yielded the currect recurrence. FindRE and MinimalRecurrence prove the recurrence, while listtorec in FindRec does not. The initial terms are only needed for MinimalRecurrence (and for the computation of seq(a(n) ...).

#16 by Peter Luschny at Fri Nov 04 11:11:45 EDT 2022

STATUS

proposed

editing

#15 by Peter Luschny at Fri Nov 04 10:16:37 EDT 2022

STATUS

editing

proposed

Discussion

Fri Nov 04

10:42

Peter Luschny: When I wwrote in A174403: "In this respect your preconditions (ogf + init) are redundant and rather error-prone." I did not yet know that it already happened here. Georg, you should reconsider your method.

11:11

Peter Luschny: I am now setting this to 'edit' since the ogf has been changed.

#14 by Peter Luschny at Fri Nov 04 10:13:23 EDT 2022

MAPLE

# Alternative, using function FindSeq from A174403:

ogf := -(10*x + sqrt((1-10*x)*(1-14*x)))/(2*x):

a := FindSeq(ogf): seq(a(n), n=0..17); # Peter Luschny, Nov 04 2022

Discussion

Fri Nov 04

10:15

Peter Luschny: That's all we need: a correct ogf and the function FindSeq. For the writer it is easier to handle and for the reader everything becomes much clearer.

#13 by Peter Luschny at Fri Nov 04 10:08:52 EDT 2022

NAME

Expansion of (1 - (10*x + sqrt((1-10*x)*(1-14*x)))/(2*x).

STATUS

proposed

editing

Discussion

Fri Nov 04

10:09

Peter Luschny: The ogf was wrong!

#12 by Jon E. Schoenfield at Fri Nov 04 01:19:09 EDT 2022

STATUS

editing

proposed

Discussion

Fri Nov 04

05:29

Peter Luschny: Why do you use functions that are only available in Maple 2022? The same can be done with gfun, which is also included in older versions.

10:05

Peter Luschny: Ups, ogf := (1 - 10*x + sqrt((1-10*x)*(1-14*x)))/(2*x):
series(ogf, x, 26): [seq(coeff(%, x, n), n = 0..22)]; gives me -11, -1, -12, -145, -1764, ... . Georg, what happened?

#11 by Jon E. Schoenfield at Fri Nov 04 01:18:56 EDT 2022

NAME

Expansion of (1 - 10*x + sqrt((1-10*x)*(1-14*x)))/(2*x).

COMMENTS

The g.f. A(x) satisfies the continued fraction relation A(x) = 1/(1-x/(1-10*x-x*A(x))).

FORMULA

a(n) = sqrt(5/7) * 10^n * (6*hypergeom([1/2, n+1],[1],2/7)-7*hypergeom([1/2, n],[1],2/7)) / (n+1) for n > 0. [_- _Mark van Hoeij_, Jul 02 2010]

STATUS

proposed

editing

#10 by Michel Marcus at Thu Nov 03 18:19:52 EDT 2022

STATUS

editing

proposed