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

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

ISBN-10: 3540111735

ISBN-13: 9783540111733

ISBN-10: 3540389539

ISBN-13: 9783540389538

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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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3. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. , Addison-Wesley, Reading, 1976. 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.

3), we deduce the desired result. Appell [3], Broden [9], [ I 0 ] , and Picard [27] have studied periodic solutions of f(x+h)+pf(x) = r For a discussion of solutions to many types of difference equations and for numerous references, see Norlund's a r t i c l e in the Encyklopadie [25]. ENTRY 4. 1) (p+l)-n-lCn(p) = ~ (k+l)n(-p) k. k=O PROOF. = ~ e - ( k + l ) X ( _ p ) k eX+p k=O : ~ n=O k=O I-X! p) k. 1). 2) have a common domain of convergence in the complex x-plane. -~ < m <_ ~. 3) converges i f and only i f Ixl 2 < (Log r)2+~2.

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. For Ixl < ILog(-P)l, )n~2n+l(P)x2n+l sin x : Z (-I p2+2p cos x + 1 n=O ( 2 n + l ) ~ ( p + l ) 2n+2 4 AND 5. 3), we have = ~ Cn(p)(-ix)n p2+2pcos x + l n=O n~(p+l) n+l Equating real and imaginary parts on both sides above, we deduce Corollaries 4 and 5, respectively. In the hypothesis of Corollary 6, which consists of four parts, Ramanujan attempts to define a sequence of numbers {An}.

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Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980 by Emil Grosswald (auth.), Marvin I. Knopp (eds.)

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