Analytic Number Theory: Proceedings of a Conference Held at by Emil Grosswald (auth.), Marvin I. Knopp (eds.)

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

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Extra info for Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980

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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. 12. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, London, 1935.

63 EXAMPLE 2. If eP+Q+R+S = eP+eQ+eR+eS-2 e-P+e-Q+e-R+e-S_2 , then 1 PROOF. The given equality is equivalent (1+p)(1+q)(l+r)(l+s) to : 2+p+q+r+s 2-p-q-r-s" Now cross multiply and ignore all products involving p2, q2, r 2, and s2. a tedious calculation and much simplification, -2 : l + l + ! + ! p Proceeding q r After we find that s " as in Example I, we achieve the sought approximation. EXAMPLE 3. If 2eP+Q+R+S+T : {eP+eQ+eR+eS+eT-2) 2- {e2P+e2Q+e2R+e2S+e2T-2) e-P+e-Q+e-R+e-S+e-T_2 then 1 :I PROOF.

1), i f IPl < I, n kZl: kn(-p)k-I = j=lZ FJ(n)(-P)J-I k=O Z (n~k)4-P)k Equating the coefficients of pr-I on both sides, we deduce ( i i ) . PROOF OF ( l i i ) . 1), for IPl < I, n-I n~l [n+l kZO Fk+l(n)(-P)k = ~ (k+l)n(-P) k )pJ k=O j=O = " J " Equate coefficients of pr-I on both sides to deduce 4 i i i ) . The statement of ( i i ) in the notebooks [31, vol. II, p. 48] is incorrect; replace n by n+l on the l e f t side of ( i l ) in [31]. zky [38] while ( l i l ) ENTRY 7. 1) Entry 6(11) is due to Worpit- is due to Euler [19].

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