By Emil Grosswald (auth.), Marvin I. Knopp (eds.)
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Additional resources for Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980
L (1950), 194-198. II. R. Bellman, A Brief Introduction to Theta Functions, Holt, Rinehart and Winston, New York, 1961. 12. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, London, 1935. 13. A. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Amer. J. , 72 (1952), 474-500. 14. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomontl, fratrum Borntrager, 1829 (Reprinted: in vol. I, 49-239 of Jacobi's, Gesammelte Werke, Reimer, Berlin 1881-1891, now by Chelsea, New York, 1969).
Note that E = eD. We shall f i r s t derive another formulation of F2n+l(X). From ( 2 . 4) F2n+l(X) : ~ (-l)k(n~) 2Eke k=-n (n+k)' (n-k) ,. 5) (-l)n(n~)Z Z (2n+l)' n=O Set y = sinb(D/2) = ( ~ - I / ~ ) / 2 . (2 sinh(D/2) )2n+Ir : (Note that (vrE-I/~)nr 2D~ eDl2+e_b/2. ) calculation shows that /y2~l = CeO/2+e-D/2)/2. 6) is valid for y = O. 6) are equal. 7), we easily obtain after a short calculation. =r ~2n). " r This completes the proof. 4), we find that the l e f t side of (2,31 may be written as (-2 sinh2(D/2}}nr = n=O 1 _ I+2 sinh2(D/2) l r r cosh D Since the given difference equation in operator notation is (E+E-l)f : 2r f : r D, the desired result follows.
Q'2N)2N(q),(q2N+l). : (• (_l)n+Nqn2+2nN+N2(q;q2)n+N' N2+N n=O (• (q2;q2)n(q2;q2)n+2 N N(2N+I] (q2;q2). (• (q2;q2)2N (_l)nqn2+2nN(q2N+l q2) n n=O (q2;q2)n(q4N+2;q2)n (~iq-N)q2N2+N(q;q2)~ (q2;q2). (• (•174 (q2;q2). (q2;q2)~ CASE3. e. a2 § -I). (l+a2)f(a) : (t~) | (-l)nqn2(q;q2)n Z n=0 (q2;q2)~ 1 (~i)(q;q2)- ~ (q2;q2)| 1 (q),~(q;q2)| (• ~- (_q) (q)2 :0. 40 (• (q2;q2)~ 1 (• 2 (q2;q2), 1 2 (• (q2;q2)~ (by [3; p. 20, Col. 4]) =0. 11) proved valid. 12). 11) above) is an immediate corollary of the q-analog of Gauss's theorem [3; p.