By Emil Grosswald (auth.), Marvin I. Knopp (eds.)

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4. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly, 86 C1979), 89-I08, 5. E. Andrews, Ramanujan and his "lost" notebook, Vinculum, 16 (1979), 91-94. 6. E. Andrews, Partitions: Yesterdayand Today, New Zealand Math. , Wellington, 1979. 7. E. , {to appear). 8. E. , (to appear). 47 9. E. , (to appear). I0. N. Bailey, On the basic bilateral hypergeometric series 2~2, Quart. J. , l (1950), 194-198. II. R. Bellman, A Brief Introduction to Theta Functions, Holt, Rinehart and Winston, New York, 1961.

Since the proofs use operator calculus, we need to define a couple of operators. As customary, l e t D denote the differential operator and define E by Ef(x~ = fCx+l). PROOF OF ENTRY2 ( i ) . 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.

J=O COROLLARY I. Let f(x) denote the solution found in Entry 3. 2) 4x)/nk enh+p where i t is understood that the series in the numerator above does, indeed, converge. PROOF. 3). 2), the proposed result readily follows. Corollary 2 is a complete t r i v i a l i t y and not worth recording here. COROLLARY 3. If n is even and positive, then Cn(p) is divisible by l-p. PROOF. 1) and Entry 6(i). 59 COROLLARY 4. For Ixl < [LogC-p}I, cos x + p : Z (-l)n~2n(P)x2n p2+2p cos x + 1 n=O (2n)~(p+l) 2n+l COROLLARY 5.

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