(Let's try this with Dr. Stay's BadRapBane. I hope there was plenty of blood spilled at the the GMail design meetings where the current behavior was defaulted.)

Weird. The Courier font went way tiny.

Theorem 4.5 in Bruce's http://www.math.uiuc.edu/~berndt/articles/rrcf.pdf indeed reduces the problem of evaluating the four Pochhammers in f(q)/g(q) to two ratios of "second differences" of the form eta(q^2)^2/eta(q)/eta(q^4), q:=e^(pi alpha). These are even easier to evaluate than plain etas--their gamma components have been divided out, and they have a lower algebraic order.

This cleverly dodges the problem of evaluating naked (q;q)_oo, let alone f and g, which are of the form (q^r,q^(1-r);q)_oo for rational r. I can get these (even quadratic surd r?) by guessing plus lattice reduction, but is there a theory? And does the theory preclude closed forms for naked (q^r;q)_oo, 2r noninteger? It resembles the problem of evaluating Gamma(r). --rwg

Ah, here's what I was after--an expression of theta_1(pi/5,q) in terms of theta constants:

EllipticTheta[1, \[Pi]/5, E^(\[Pi]^2/(5 Log[q]))]==(Sqrt[5] r^( 1/4) (11 - 11 r + 5 r^2 - r^3 + (5 + (-4 + r) r) Sqrt[5 + (-2 + r) r])^(1/10) EllipticTheta[4, 0, q^5] Sqrt[-Log[q]])/(2^(1/10) Sqrt[\[Pi]] (-1 + r)^(3/5))

(after an unbelievable struggle against FullSimplify, which can't even confirm algebraic equivalence with my starting expression) with

{Rule[s, Sqrt[r^3 - 2*r^2 + 5*r]], Rule[r, ((EllipticTheta[2, 0, q]* EllipticTheta[3, 0, q])/(EllipticTheta[2, 0, q^5]* EllipticTheta[3, 0, q^5]))]}

This reduces evaluating the R-R CF to evaluating etas. Theoretically, just two, since, if r is rational, eta(q^r) is a (probably brutal) algebraic function of eta(q) and eta(q^2). e.g., 256*eta[q]^2*eta[q^10]^24 == 5*eta[q^5]^18*eta[q^10]^8 + 10*eta[q]^6*eta[q^5]^12*eta[q^10]^8 + eta[q]^12*eta[q^5]^6*eta[q^10]^8 - eta[q]^2*eta[q^5]^24 . (This is a *tiny* one.)

Mma's woeful Theta machinery was of no help on this, except to speed up some of the failed brute force approaches. The solution was in my old q-Trig paper, whose abstract prophesied:

We introduce a q-generalization of the sine and cosine functions, related to the θ functions, but (as revealed by computer experiments) possessing addition and multiplication formulas more analogous to those of ordinary sin and cos. These formulas then contribute identities to θ theory, and hint of a more natural formulation of θ functions as outgrowths of elementary functions. Nevertheless, this paper can be read without knowledge of θ functions--it was certainly written that way.

The paper was soundly rejected by the august SYMSAC '76 Computer Algebra seance, but later won high praise from Dyson.

Three by-products: Equal[EllipticTheta[1, Pi/10, E^((Pi^2)/(5*Log[q]))], ((EllipticTheta[4, 0, q^5]*Sqrt[5]* Sqrt[-Log[q]]*(s*Sqrt[s^2 - 4*r] - s^2 + 2*r + 2)^(1/5))/(2^(1/ 5)*(r - 1)^(1/5)*(s^2 - r^2 - 2*r - 1)^(1/5)*Sqrt[Pi]))]

Equal[ EllipticTheta[2, Pi/10, E^((Pi^2)/(5*Log[q]))], ((EllipticTheta[4, 0, q^5]*Sqrt[5]* Sqrt[-Log[q]]*(s*Sqrt[s^2 - 4*r] - s^2 + 2*r + 2)^(1/5)* Sqrt[((s*Sqrt[s^2 - 4*r] + (1 - 2*r)*s^2 + 2*r^3 + 2*r^2)/((s^2 - r^2 - r)*Sqrt[s^2 - 4*r] - s^3 + (r^2 + r + 2)*s))])/(2^(1/5)*(r - 1)^(1/5)*(s^2 - r^2 - 2*r - 1)^(1/5)*Sqrt[Pi]))]

Equal[ EllipticTheta[2, Pi/5, E^((Pi^2)/(5*Log[q]))], ((2*EllipticTheta[4, 0, q^5]*Sqrt[5]* Sqrt[-Log[ q]]*((-s^4 + (2*r^2 + 3*r)*s^2 - r^4 - 2*r^3 - r^2)* Sqrt[s^2 - 4*r] + s^5 + (-2*r^2 - 5*r)*s^3 + (r^4 + 6*r^3 + 5*r^2)*s)^(3/ 5))/((s^2 - r^2 - 2*r - 1)^(1/ 5)*((r - 1)^(1/5)*(2^(3/5)*s^2 - 2^(3/5)*r^2 - 2^(3/5)*r)* Sqrt[s^2 - 4*r] + (r - 1)^(1/5)*((2^(3/5)*r^2 + 3*2^(3/5)*r)*s - 2^(3/5)*s^3))*Sqrt[Pi]))]

Besides its more serious failing on [EllipticTheta[2, Pi, q] , 0 < q < 1], FullSimplify mysteriously converts Sqrt[-5*Loq[q]] to Sqrt[Log[1/q^5]] . --rwg

On Fri, Jun 11, 2010 at 3:55 PM, <berndt@math.uiuc.edu> wrote:

...

In his notebooks and lost notebook, Ramanujan records many special values for the Rogers-Ramanujan continued fraction. Perhaps the best source for Ramanujan's contributions is Chapter 2 of Ramanujan's Lost Notebook, Part I, by George Andrews and myself. In this chapter you will find a general formula that Ramanujan might have used as well as proofs of all the values of the c.f. found in his lost notebook. In a paper I published with Heng Huat Chan and Liang-Cheng Zhang in Crelle in 1996, we presented a general method for evaluating the Rogers-Ramanujan continued fraction. We also prove a theorem showing that the values at a certain class of arguments containing all those mentioned above, up to a possible small fractional power of 2, are units in the algebraic number fields in which they lie. My paper with Chan and Zhang as well as some other papers on the Rogers-Ramanujan continued fraction are available at my web-site.

Best wishes, Bruce Berndt

...

I hadn't realized how directly connected they are to the Rogers-Ramanujan identities: If f(q):=(q,q^4;q^5)_oo and g(g):=(q^2,q^3;q^5)_OO, the reciprocals of the g.f.s for the +-1 (resp +-2) mod 5 partitions, then 5 5 f(q) 1 2 (q ;q )_oo ---- = -------- = f(q) --------- g(q) q (q;q)_oo 1 + -------- 2 q 1 + ------- 3 q 1 + ----- ...

which appears to come out an <algebraic #>/q^(1/5) when q = e^(pi r), r a quadratic surd (in the left halfplane) "by an appeal to the theory of elliptic theta functions" (G. Andrews: The Theory of Partitions, p105 :-). Ramanujan sent Hardy the cases q = e^(-2 pi) and - e^-pi, which presumably give the nicest algebraics,

f(e^-2 pi) pi 3 pi 2 pi/5 ---------- = 4 sin -- sin ---- e , g(e^-2 pi) 20 20

whereas the nicest cf is presumably

f(e^-pi) sqrt(40 + 18 sqrt(5) - 25 5^(1/4) - 11 5^(3/4)) - sqrt(36 + 18 sqrt(5) - 25 5^(1/4) - 11 5^(3/4)) pi/5 -------- = ------------------------------ ------------------------------

---

...

e . g(e^-pi) 2

~ 0.958650181596.

Now we know why Ramanujan didn't send Hardy this one.

I'm really interested in how to get these theta special values (other than by numerical searching), but Andrews seems to cite only combinatoric references rather than (theta) number-theoretic. In particular, where do I learn that f and g can be evaluated separately, e.g. here in terms of e^(pi/60) because

-pi -pi - pi/6 10 f(e ) g(e ) = e sqrt ------------------ . 5^3/4 - 5^(1/4) +2

Other forms of f and g are n (5n-3) n/2 Sum (-) q 1 Theta_4(7 i log(q)/4, q^(5/2)) n f(q) = ----------- = - q ------------------------------ = -------------------- , n^2 (q^5;q^5)_oo (q^5;q^5)_oo q Sum ------- n>0 (q;q)_n

the "gofigurate number theorem", and n (5n-1) n/2 Sum (-) q 1 2 Theta_4(9 i log(q)/4, q^(5/2)) n g(q) = ------------ = - q ------------------------------ = -------------------- . (n+1) n (q^5;q^5)_oo (q^5;q^5)_oo q Sum ------- n>0 (q;q)_n

f(q) and g(q) become Theta_1(pi/5, Q) under Jacobi's transformation. If we can express these in terms of Theta_s(0,p) ("Theta constants") we'd have a nearly automatic source of special values because the Theta constants all come out as etas, at which we're pretty good. I can do Theta_1(pi/10, Q), but so far no luck with pi/5. --rwg

-- Bruce Berndt Dept. of Mathematics University of Illinois 1409 West Green St. Urbana, IL 61801 phone: 217-333-3970 (office) fax: 217-333-9576

My web-page was updated on August 28, 2009. http://www.math.uiuc.edu/~berndt/ Please inform me of any mistakes or problems.

My favorite composers (in no particular order): Bach, Purcell, Vaughan Williams, Sibelius, Ravel, Schubert