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