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.)

ISBN-10: 3540111735

ISBN-13: 9783540111733

ISBN-10: 3540389539

ISBN-13: 9783540389538

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

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1)]). 24) as q(q8;q8) (_q4;q4). (q 2 ;q 4 )=(_q 2 4";q 4 )=2 = (_q2q4)... (q4;q4)= 4 42 (_q2;q4)= (( (q4;q4)= (_q2;q4)~ (q4;q4). 2 44 ((q ;q ) = ( - q ;q )= + 2q(q8;q8)=(-l;q4)= + (q4;q8) (q2;q4)= 4q(q4 q4) (_q4;q4) ) ~ q2n2)2 . = (by [22; p. 269, eq. (4), 5th line]) : ( ~ qn2)2/( ~ (_q2)n(n+l)/2) n=-= n=O (by [3; p. 23, eq. 10)) is proved. 11). 11), we must find a suitable representation of this function. To effect this we consider the q-analog of Whipple's theorem [21; p. lO0, eq. e. g + =), e = -f = ql/2, d = c-l and a ~ I.

Slater, Generalized Hypergeometric Functions, Cambridge University Press, London, 1966. 48 22. J. Tannery and J. Molk, Elements de la Theorie des Fonctlons Elliptiques, Vol. I I . , Gauthier-Villars, Paris, 1896 (Reprinted: Chelsea, New York, 1972). 23. J. Tannery and J. Molk, El~nents de la Theorie des Fonctions Elliptiques, Vol. I I I , Gauthier-Villars, Paris 1898 (Reprinted: Chelsea, New York, 1972). 24. N. Watson, The final problem: an account of the mock theta functions, J. London Math. , II (1936), 55-80.

For Log(sec x) = Z n=l PROOF. For Ixl < ~/2, (-l)n22n (l-22n}B2nx2n (2n)(2n)' Ixl < ~/2, Log(sec x) = fx tan t dt. 0 Now employ Entry 15. Ramanujan now makes three remarks, the f l r s t of which is trivial and the second of which is a special case of the third, B2n B2n-2h (-l)h(2n)~ (2x)2h(2n-2h)~ 85 as n tends to | where 0 5 h ~ n-1. This asymptotic formula is a simple conse- quence of Euler's formula for r see Entry 25(i). The following recurrence relation is due to Euler [19]. ENTRY 18. Let n be an integer exceeding I.

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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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