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Note on the truncated generalizations of Gauss’s square exponent theorem. (English) Zbl 1454.11044

Gauss’ Square Exponent Theorem asserts that \(\prod_{n\geq 1} \frac{1-q^n}{1+q^n}=\sum_{k=-\infty}^{\infty} (-1)^kq^{k^2}\). Much work has been done to find truncated versions of this identity. The paper under review finds new truncated versions of this identity. Some multiple summation identities are verified through computer assisted proof using Mathematica.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
05A30 \(q\)-calculus and related topics

Citations:

Zbl 1434.11055

Software:

qMultiSum
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Full Text: DOI arXiv Euclid

References:

[1] G.E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, 1998. · Zbl 0996.11002
[2] G.E. Andrews and M. Merca, The truncated pentagonal number theorem, J. Combin. Th. 119 (2012), 1639–1643. · Zbl 1246.05014
[3] ——–, Truncated theta series and a problem of Guo and Zeng, J. Combin. Th. 154 (2018), 610–619. · Zbl 1454.05015
[4] A. Berkovich and F.G. Garvan, Some observations on Dyson’s new symmetries of partitions, J. Combin. Th. 100 (2002), 61–93. · Zbl 1016.05003
[5] S.H. Chan, T.P.N. Ho and R. Mao, Truncated series from the quintuple product identity, J. Num. Th. 169 (2016), 420–438. · Zbl 1409.11017
[6] S.B. Ekhad and D. Zeilberger, The number of solutions of \(X^2=0\) in triangular matrices over \(GF(q)\), Electr. J. Combin. 3 (1996). · Zbl 0851.15010
[7] V.J.W. Guo and J. Zeng, Multiple extensions of a finite Euler’s pentagonal number theorem and the Lucas formulas, Discr. Math. 308 (2008), 4069–4078. · Zbl 1156.05003
[8] ——–, Two truncated identities of Gauss, J. Combin. Th. 120 (2013), 700–707. · Zbl 1259.05020
[9] T.Y. He, K.Q. Ji and W.J.T. Zang, Bilateral truncated Jacobi’s identity, European J. Combin. 51 (2016), 255–267. · Zbl 1321.05013
[10] F. Jouhet, Shifted versions of the Bailey and well-poised Bailey lemmas, Ramanujan J. 23 (2010), 315–333. · Zbl 1204.33021
[11] L.W. Kolitsch, Another approach to the truncated pentagonal number theorem, Int. J. Num. Th. 11 (2015), 1563–1569. · Zbl 1327.11071
[12] J.C. Liu, Some finite generalizations of Euler’s pentagonal number theorem, Czechoslovak Math. J. 67 (2017), 525–531. · Zbl 1458.05025
[13] ——–, Some finite generalizations of Gauss’s square exponent identity, Rocky Mountain J. Math. 47 (2017), 2723–2730. · Zbl 1434.11055
[14] R. Mao, Proofs of two conjectures on truncated series, J. Combin. Th. 130 (2015), 15–25. · Zbl 1316.11092
[15] A. Riese, qMultiSum–A package for proving \(q\)-hypergeometric multiple summation identities, J. Symbol. Comp. 35 (2003), 349–376. · Zbl 1020.33007
[16] S.O. Warnaar, \(q\)-Hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem, Ramanujan J. 8 (2004), 467–474. · Zbl 1066.05023
[17] A.J. Yee, A truncated Jacobi triple product theorem, J. Combin. Th. 130 (2015), 1–14. · Zbl 1379.11027
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