[seqfan] Re: Some conjectured relations in the database (involving Ramanujan's functions)

[Neil Sloane]

Thomas, That's very interesting!

The thing to do would be to add a line like this to the FORMULA section of
each of the sequence involved, saying something like

It appears that  -3*A138521(n) + 15*A138520(n) - 16*A213265(n) +
4*A261321(n) = 0. - ~~~~

where of course the four tildes at the end get replaced with your name and
date.

But you realize that by focusing on Ramanujan, you have entered into a
field, elliptic functions and theta series, where there are
an incredibly huge number of known identities, going back hundreds of years
- see Tannery and Molk's 1893 volumes (available from Chelsea).
Even specialists in the field have been known to refer to the "morass" of
such identities.

Michael Somos (one of the OEIS editors, and formerly on this list - and may
still be) is an expert.

So your Ramanujan-based identities may well be new, and should be put on
record somewhere (perhaps create a page on the OEIS Wiki where you can list
them all).  But you might find more gold nuggets
if you picked a field which is newer.  Any branch of combinatorics, for
instance. Graph theory, say.




Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Sat, Dec 16, 2017 at 1:04 PM, Thomas Baruchel <baruchel at gmx.com> wrote:

> Dear fellow seqfans,
>
> a new attempt in detecting relations between sequences from the database
> by using computing means resulted this time in a list of linearly dependant
> sequences. In order to focus on a set of ~1000 interesting sequences where
> such relations could be found, I have been working on those having the name
> of Ramanujan in their main title (in the downloadable file names.gz).
>
> I didn't spend time looking deeper in the full articles from the database
> in order to see whether the following identities/conjectures were already
> reported or not; please feel free to add them if you think it may have some
> interest. On my own side, I will rather tune my algorithms with new
> settings
> and explore some new ideas. If by chance the following identities bring
> something new, I can yield more of them; feel free to ask also.
>
> The most elaborated conjecture in my list involves four sequences :
>
>   -3*A138521(n) + 15*A138520(n) - 16*A213265(n) + 4*A261321(n) = 0
>
> In order to have a normalized way of writing all the identities, please
> consider
> the following output for the same one:
>
>  ['A138521', 'A138520', 'A213265', 'A261321'] --> [-3, 15, -16, 4] (506)
> A138520 Expansion of 1 - q * (psi(q^5) / psi(q))^2 in powers of q where
> psi() is a Ramanujan theta function.
> A138521 Expansion of chi(-q)^5 / chi(-q^5) in powers of q where chi() is a
> Ramanujan theta function.
> A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers
> of q where psi() is a Ramanujan theta function.
> A261321 Expansion of (phi(q) / phi(q^3))^2 in powers of q where phi() is a
> Ramanujan theta function.
>
> Thus you can read:
>
>   * the list of involved sequences;
>   * the vector detected by my program;
>   * the norm of the vector (between parentheses);
>   * the name of all involved sequences.
>
>
> All other conjectures involve three sequences:
>
>  ['A233693', 'A132975', 'A128143'] --> [2, -3, 1] (14)
> A128143 Expansion of q* (psi(q^9)/phi(q^9))/ (psi(q)/phi(q)) in powers of
> q where psi(),phi() are Ramanujan theta functions.
> A132975 Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is
> a Ramanujan theta function.
> A233693 Expansion of q * psi(-q) * chi(-q^6) * psi(-q^9) / (phi(-q) *
> phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta
> functions.
>
>  ['A244544', 'A244543', 'A244540'] --> [-2, 1, 1] (6)
> A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where
> phi() is a Ramanujan theta function.
> A244543 Expansion of phi(q^2) * (phi(q) + phi(q^2)) / 2 in powers of q
> where phi() is a Ramanujan theta function.
> A244544 Expansion of (phi(q) + phi(q^2))^2 / 4 in powers of q where phi()
> is a Ramanujan theta function.
>
>  ['A112128', 'A208604', 'A232358'] --> [2, -1, -1] (6)
> A112128 Expansion of phi(q^4) / phi(q) in powers of q where phi() is a
> Ramanujan theta function.
> A208604 Expansion of phi(-q) / phi(q^4) in powers of q where phi() is a
> Ramanujan theta function.
> A232358 Expansion of phi(q^2)^2 / (phi(q) * phi(q^4)) in powers of q where
> phi() is a Ramanujan theta function.
>
>  ['A260301', 'A257656', 'A266684'] --> [1, -2, 1] (6)
> A257656 Expansion of f(x) * f(x^3) * f(-x^4)^2 * chi(-x^6)^2 in powers of
> x where chi(), f() are Ramanujan theta functions.
> A260301 Expansion of f(-x^3)^3 * psi(x)^3 / psi(x^3)^2 in powers of x
> where phi(), f() are Ramanujan theta functions.
> A266684 Expansion of f(-x) * f(-x^2)^4 / psi(x^3) in powers of x where
> psi(), f() are Ramanujan theta functions.
>
>  ['A215346', 'A233458', 'A131124'] --> [2, -1, -1] (6)
> A131124 Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where
> phi(), psi() are Ramanujan theta functions.
> A215346 Expansion of (1/q) * phi(-q) * phi(q^4) / (phi(q) * psi(q^8)) in
> powers of q where phi(), psi() are Ramanujan theta functions.
> A233458 Expansion of q^(-1) * (phi(q^2) * phi(-q) / psi(-q^2)^2)^2 in
> powers of q where phi(), psi() are Ramanujan theta functions.
>
>  ['A258034', 'A256280', 'A258256'] --> [3, 1, -4] (26)
> A256280 Expansion of phi(q^3)^4 / (phi(q) * phi(q^9)) in powers of q where
> phi() is a Ramanujan theta function.
> A258034 Expansion of phi(q) * phi(q^9) in powers of q where phi() is a
> Ramanujan theta function.
> A258256 Expansion of f(q^3) * psi(-q^3)^3 / (psi(-q) * psi(-q^9)) in
> powers of q where psi(), f() are Ramanujan theta functions.
>
>  ['A246926', 'A213056', 'A259884'] --> [1, 1, -2] (6)
> A213056 Expansion of chi(x) * f(x^3)^3 in powers of x where chi(), f() are
> Ramanujan theta functions.
> A246926 Expansion of phi(x)^2 * chi(x) * psi(-x^3) in powers of x where
> phi(), psi(), chi() are Ramanujan theta functions.
> A259884 Expansion of phi(x) * f(-x^3)^3 / f(-x) in powers of x where
> phi(), f() are Ramanujan theta functions.
>
>  ['A257657', 'A260412', 'A132969'] --> [-1, 2, -1] (6)
> A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are
> Ramanujan theta functions.
> A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x
> where f(,) is Ramanujan's general theta function.
> A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x
> where psi(), f(,) are Ramanujan theta functions.
>
>  ['A247223', 'A133985', 'A121373'] --> [2, -1, -1] (6)
> A121373 Expansion of f(x) = f(x, -x^2) in powers of x where f(, ) is
> Ramanujan's general theta function.
> A133985 Expansion of f(-x, x^2) in powers of x where f(, ) is Ramanujan's
> general theta function.
> A247223 Expansion of f(-x^5, -x^7) in powers of x where f() is a Ramanujan
> theta function.
>
>  ['A258322', 'A256014', 'A256282'] --> [3, 1, -4] (26)
> A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q
> where phi() is a Ramanujan theta function.
> A256282 Expansion of f(-q^3) * psi(q^3)^3 / (psi(q) * psi(q^9)) in powers
> of q where psi(), f() are Ramanujan theta functions.
> A258322 Expansion of phi(-q) * phi(-q^9) in powers of q where phi() is a
> Ramanujan theta function.
>
>  ['A260412', 'A257657', 'A260221'] --> [-2, -1, 3] (14)
> A257657 Expansion of f(-x, -x) * f(-x^6, -x^6) / f(x, x^2) in powers of x
> where f(,) is Ramanujan's general theta function.
> A260221 Expansion of phi(x^3)^2 / f(x) in powers of x where phi(), f() are
> Ramanujan theta functions.
> A260412 Expansion of psi(x^2) * psi(x^3) / f(-x^2, -x^10) in powers of x
> where psi(), f(,) are Ramanujan theta functions.
>
>  ['A261326', 'A213265', 'A233672'] --> [1, 2, -3] (14)
> A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers
> of q where psi() is a Ramanujan theta function.
> A233672 Expansion of psi(q) * phi(-q^18) * f(-q^6) / f(q^3)^3 in powers of
> q where phi(), psi(), f() are Ramanujan theta functions.
> A261326 Expansion of f(-x^2, -x^4)^2 / (f(x^3, -x^6) * f(-x, x^2)) in
> powers of x where f(,) is Ramanujan's general theta function.
>
>  ['A115978', 'A253626', 'A253623'] --> [1, -2, 1] (6)
> A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a
> Ramanujan theta function.
> A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q
> where phi(), f() are Ramanujan theta functions.
> A253626 Expansion of psi(q^2) * f(q, q^2)^2 / f(q, q^5) in powers of q
> where psi(), f() are Ramanujan theta functions.
>
>  ['A131124', 'A233458', 'A215346'] --> [1, 1, -2] (6)
> A131124 Expansion of q^(-1) * (phi(-q) / psi(q^4))^2 in powers of q where
> phi(), psi() are Ramanujan theta functions.
> A215346 Expansion of (1/q) * phi(-q) * phi(q^4) / (phi(q) * psi(q^8)) in
> powers of q where phi(), psi() are Ramanujan theta functions.
> A233458 Expansion of q^(-1) * (phi(q^2) * phi(-q) / psi(-q^2)^2)^2 in
> powers of q where phi(), psi() are Ramanujan theta functions.
>
>  ['A138949', 'A122857', 'A125061'] --> [1, 3, -4] (26)
> A122857 Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where
> phi() is a Ramanujan theta function.
> A125061 Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of
> q where psi(), chi() are Ramanujan theta functions.
> A138949 Expansion of (3 * phi(q^3)^2 - phi(q)^2) / 2 in powers of q where
> phi() is a Ramanujan theta function.
>
>  ['A121455', 'A133691', 'A109506'] --> [-1, -1, 2] (6)
> A109506 Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a
> Ramanujan theta function.
> A121455 Expansion of q*(phi(-q)psi(q^4))^2 in powers of q where phi(),
> psi() are Ramanujan theta functions.
> A133691 Expansion of (1 - (phi(-q) * phi(q^2))^2) / 4 in powers of q where
> phi() is a Ramanujan theta function.
>
>  ['A233672', 'A213265', 'A261326'] --> [3, -2, -1] (14)
> A213265 Expansion of psi(q) * psi(q^2) * psi(q^6) / psi(q^3)^3 in powers
> of q where psi() is a Ramanujan theta function.
> A233672 Expansion of psi(q) * phi(-q^18) * f(-q^6) / f(q^3)^3 in powers of
> q where phi(), psi(), f() are Ramanujan theta functions.
> A261326 Expansion of f(-x^2, -x^4)^2 / (f(x^3, -x^6) * f(-x, x^2)) in
> powers of x where f(,) is Ramanujan's general theta function.
>
>  ['A261154', 'A132972', 'A128144'] --> [2, -1, -1] (6)
> A128144 Expansion of chi(-q)* chi(-q^2)* chi(-q^9)/( chi(-q^3)* chi(q^9))
> in powers of q where chi() is a Ramanujan theta function.
> A132972 Expansion of chi(q)^3 / chi(q^3) in powers of q where chi() is a
> Ramanujan theta function.
> A261154 Expansion of psi(q^6) * f(-q^12) / (psi(-q) * psi(q^9)) in powers
> of q where psi(), f() are Ramanujan theta functions.
>
>  ['A133985', 'A247223', 'A121373'] --> [-1, 2, -1] (6)
> A121373 Expansion of f(x) = f(x, -x^2) in powers of x where f(, ) is
> Ramanujan's general theta function.
> A133985 Expansion of f(-x, x^2) in powers of x where f(, ) is Ramanujan's
> general theta function.
> A247223 Expansion of f(-x^5, -x^7) in powers of x where f() is a Ramanujan
> theta function.
>
>
> Best regards,
>
> --
> th. baruchel
>
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>