Zagier,s Identity
المؤلف:
Andrews, G. E.
المصدر:
"Two Theorems of Gauss and Allied Identities Proved Arithmetically." Pacific J. Math. 41,
الجزء والصفحة:
...
2-9-2019
2641
Zagier's Identity
(OEIS A117586; Andrews 1972, 1998; Knuth and Paterson 1978; Chapman 2000; Zagier 2001), where
![g(q)=sum_(n=1)^infty(-1)^n[(3n-1)q^(n(3n-1)/2)+3nq^(n(3n+1)/2)],](http://mathworld.wolfram.com/images/equations/ZagiersIdentity/NumberedEquation1.gif) |
(5)
|
is a Lambert series, and 
is a q-polygamma function.
If
![f(x,q)=1+sum_(n=1)^infty(-1)^n[x^(3n-1)q^(n(3n-1)/2)+x^(3n)q^(n(3n+1)/2)],](http://mathworld.wolfram.com/images/equations/ZagiersIdentity/NumberedEquation2.gif) |
(6)
|
then a related identity is given by
(Subbarao 1971, Andrews 1972, Andrews 1983, Knuth and Paterson 1978, Chapman 2000, Zagier 2001).
REFERENCES:
Andrews, G. E. "Two Theorems of Gauss and Allied Identities Proved Arithmetically." Pacific J. Math. 41, 563-578, 1972.
Andrews, G. E. "Euler's Pentagonal Number Theorem." Math. Mag. 56, 279-284, 1983.
Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, 1998.
Chapman, R. "Franklin's Argument Proves an Identity of Zagier." Electronic J. Combinatorics 7, No. 1, R54, 1-5, 2000. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r54.html.
Knuth, D. E. and Paterson, M. S. "Identities from Partition Involutions." Fib. Quart. 16, 198-212, 1978.
Sloane, N. J. A. Sequence A117586 in "The On-Line Encyclopedia of Integer Sequences."
Subbarao, M. V. "Combinatorial Proofs of Some Identities." Proc. Washington State University Conference on Number Theory. Washington State University, pp. 80-91, 1971.
Zagier, D. "Vassiliev Invariants and a Strange Identity Related to the Dedekind Eta-Function." Topology 40, 945-960, 2001.
الاكثر قراءة في التفاضل و التكامل
اخر الاخبار
اخبار العتبة العباسية المقدسة
