## A polynomial identity implying Schur’s partition theorem

HTML articles powered by AMS MathViewer

- by Ali Kemal Uncu PDF
- Proc. Amer. Math. Soc.
**148**(2020), 3307-3324 Request permission

## Abstract:

We propose and prove a new polynomial identity that implies Schur’s partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kurşungöz. We also present some related polynomial and $q$-series identities.## References

- Krishnaswami Alladi, George Andrews, and Basil Gordon,
*Refinements and generalizations of Capparelli’s conjecture on partitions*, J. Algebra**174**(1995), no. 2, 636–658. MR**1334229**, DOI 10.1006/jabr.1995.1144 - Krishnaswami Alladi and Alexander Berkovich,
*A double bounded version of Schur’s partition theorem*, Combinatorica**22**(2002), no. 2, 151–168. Special issue: Paul Erdős and his mathematics. MR**1909082**, DOI 10.1007/s004930200008 - Krishnaswami Alladi and Basil Gordon,
*Generalizations of Schur’s partition theorem*, Manuscripta Math.**79**(1993), no. 2, 113–126. MR**1216769**, DOI 10.1007/BF02568332 - George E. Andrews,
*On Schur’s second partition theorem*, Glasgow Math. J.**8**(1967), 127–132. MR**220692**, DOI 10.1017/S0017089500000197 - George E. Andrews,
*On partition functions related to Schur’s second partition theorem*, Proc. Amer. Math. Soc.**19**(1968), 441–444. MR**225741**, DOI 10.1090/S0002-9939-1968-0225741-2 - George E. Andrews,
*Schur’s theorem, Capparelli’s conjecture and $q$-trinomial coefficients*, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 141–154. MR**1284057**, DOI 10.1090/conm/166/01622 - George E. Andrews,
*The theory of partitions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR**1634067** - Alexander Berkovich and Ali Kemal Uncu,
*A new companion to Capparelli’s identities*, Adv. in Appl. Math.**71**(2015), 125–137. MR**3406960**, DOI 10.1016/j.aam.2015.09.012 - Alexander Berkovich and Ali Kemal Uncu,
*Polynomial identities implying Capparelli’s partition theorems*, J. Number Theory**201**(2019), 77–107. MR**3958042**, DOI 10.1016/j.jnt.2019.02.028 - Alexander Berkovich and Ali Kemal Uncu,
*Elementary polynomial identities involving $q$-trinomial coefficients*, Ann. Comb.**23**(2019), no. 3-4, 549–560. MR**4039549**, DOI 10.1007/s00026-019-00445-8 - A. Berkovich and A. K. Uncu,
*Refined $q$-trinomial coefficients and two infinite hierarchies of $q$-series identities*, arXiv:1810.12048 [math.NT], 2018. - Christine Bessenrodt,
*A combinatorial proof of a refinement of the Andrews-Olsson partition identity*, European J. Combin.**12**(1991), no. 4, 271–276. MR**1120413**, DOI 10.1016/S0195-6698(13)80109-6 - David M. Bressoud,
*A combinatorial proof of Schur’s 1926 partition theorem*, Proc. Amer. Math. Soc.**79**(1980), no. 2, 338–340. MR**565367**, DOI 10.1090/S0002-9939-1980-0565367-X - George Gasper and Mizan Rahman,
*Basic hypergeometric series*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR**2128719**, DOI 10.1017/CBO9780511526251 - Shashank Kanade and Matthew C. Russell,
*Staircases to analytic sum-sides for many new integer partition identities of Rogers-Ramanujan type*, Electron. J. Combin.**26**(2019), no. 1, Paper No. 1.6, 33. MR**3904822** - Kağan Kurşungöz,
*Andrews-Gordon type series for Capparelli’s and Göllnitz-Gordon identities*, J. Combin. Theory Ser. A**165**(2019), 117–138. MR**3913053**, DOI 10.1016/j.jcta.2019.02.001 - Kağan Kurşungöz,
*Andrews-Gordon type series for Kanade-Russell conjectures*, Ann. Comb.**23**(2019), no. 3-4, 835–888. MR**4039566**, DOI 10.1007/s00026-019-00470-7 - K. Kurşungöz,
*Andrews-Gordon type series for Schur’s partition identity*, arXiv:1812.10039 [math.CO], 2018. - Axel Riese,
*qMultiSum—a package for proving $q$-hypergeometric multiple summation identities*, J. Symbolic Comput.**35**(2003), no. 3, 349–376. MR**1962799**, DOI 10.1016/S0747-7171(02)00138-4 - Carsten Schneider,
*Symbolic summation assists combinatorics*, Sém. Lothar. Combin.**56**(2006/07), Art. B56b, 36. MR**2317679** - Issai Schur,
*Gesammelte Abhandlungen. Band III*, Springer-Verlag, Berlin-New York, 1973 (German). Herausgegeben von Alfred Brauer und Hans Rohrbach. MR**0462893** - A. K. Uncu,
*On double sum generating functions in connection with some classical partition theorems*, arXiv:1811.08261 [math.CO], 2018. - S. Ole Warnaar,
*$q$-trinomial identities*, J. Math. Phys.**40**(1999), no. 5, 2514–2530. MR**1686533**, DOI 10.1063/1.532880

## Additional Information

**Ali Kemal Uncu**- Affiliation: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Altenbergerstrasse 69A, 4040 Linz, Austria
- MR Author ID: 1129887
- ORCID: 0000-0001-5631-6424
- Email: akuncu@risc.jku.at
- Received by editor(s): March 4, 2019
- Received by editor(s) in revised form: September 17, 2019, December 20, 2019, and December 27, 2019
- Published electronically: March 23, 2020
- Additional Notes: Research of the author was supported by the Austrian Science Fund FWF, SFB50-07, -09, and -11 Projects.
- Communicated by: Amanda Folsom
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 3307-3324 - MSC (2010): Primary 05A15, 05A17, 05A19, 11B37, 11P83
- DOI: https://doi.org/10.1090/proc/14991
- MathSciNet review: 4108840