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

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

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Read or Download Analytic Number Theory: Proceedings of a Conference Held at Temple University, Philadelphia, May 12–15, 1980 PDF

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

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4]) =0. 11) proved valid. 12). 11) above) is an immediate corollary of the q-analog of Gauss's theorem [3; p. 20, Cor. 4]. 8) once one observes that ( ~ (_l)nqn2) (q;q2)~ i012 (q)~(q;q2)~ (_q)(ql2o (q;q2), (q2;q2)~ Z qn(n+l)/2 n=O by [3; p. 23, eq. 13)]. 13). While this result is included here because of the relationship of the series therein to the other identities in this paper, i t is a direct corollary of the main theorem of [7] (see [ l ] for an elegant alternative proof). 31) (-bq)= ~ (_l)mqm 2 m:O (-q2b-l;q2) m (-I)mqm2(q ;q2)m Z m=O (-bq2;q2)m(-b-lq2;q2)m (by [18; p.

Mordell, The value of the definite integral f " eat2+btdt/(eCt+d), Quarterly Journal of Math,, 68 C1920), 329-342. 17. J. Mordell, The definite integral f " eax2+bXdx/(eCX+d) and the analytic theory of numbers, Acta Math. Stockholm, 61 (1933), 323-360. 18. S. Ramanujan, Collected Papers, Cambridge University Press, London and New York, 1927; reprinted Chelsea, New York. 19. B. Sears, On the transformation theory of basic hypergeometric functions, Proc. London Math. Soc. (2), 53 (1951), 158-180.

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. 2) are absolutely convergent when Re x > Log r. there is a common domain of convergence, and the proof is complete. ENTRY 5. 1) = Z (-l)k(~](p+l)-k~k(p). k=O PROOF. 1) holds for n > I . 1). l (p}' Note that el(p) = I. ~n(p) is a polynomial in p of degree n - l . 1) ENTRY 6. (i) (ii) Let 1 < r < n. 2) (-l)n-l~n(p) : pn-lCn41/p), n > I. 2) and equating coefficients of pr-I on both sides, we complete the proof of Ci).

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