An Analytic Proof of Some Part of Keith-Xiong’s Theorem

Ya Gao; Xinhua Xiong

doi:doi:10.11648/j.ijtam.20220801.12

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An Analytic Proof of Some Part of Keith-Xiong’s Theorem

Ya Gao, Xinhua Xiong

Published in International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 1)

Received: 8 March 2021 Accepted: 4 March 2022 Published: 15 March 2022

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Abstract

In the theory of partitions, Euler’s partition theorem involving odd parts and different parts is one of the famous theorems. It states that the number of partitions of an integer n into odd parts is equal to the number of partitions of n into different parts. By intepretting odd parts as parts congruent to 1 modulo 2, the second author and Keith provided a completely generalization about Euler’s partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan-Andrews-Gordon identities related to this theorem. They gave a combinatorial proof of the theorem by establishing bijection. In this note, we will offer an anclytic view point of this beautiful theorem. We use q-series and generating function theories to provide an analytic style proof for some cases of Keith-Xiong’s theorem. By defining basic units and special units, the basic units in the partitions are divided into two categories, and then the number between the basic units in the special units is classified, and all the cases when m = 3 and alternative sum type (Σ,2) are given, our method is verifying the generating functions of both sides satisfying the same recurrences.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 8, Issue 1)
DOI	10.11648/j.ijtam.20220801.12
Page(s)	14-29
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Partition, q−series, Generating Function

References

[1]	H. L. Alder, Partition identities - from Euler to the present, Am. Math. Monthly 76 (1969), 733-746.
[2]	G. E. Andrews, The theory of partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998.
[3]	G. E. Andrews, On generalizations of Euler’s partition theorem, Michigan Math. J. 1 (1966), 491–498.
[4]	W. Y. C. Chen, H. Y. Gao, K. Q. Ji, M. Y. X. Li, A unification of two refinements of Euler’s partition theorem, Ramanujan J. 23 (2010), 137-149.
[5]	S. Fu, D. Tang, A. J. Yee A Lecture Hall Theorem for m- Falling Partitions, Annals of Combinatorics, 23 (2019), Number 3-4, 749–764.
[6]	D. Kim and A. J. Yee, A note on partitions into distinct parts and odd parts, Ramanujan J. 3 (1999) 227–231.
[7]	I. Pak and A. Postnikov, A generalization of Sylvester’s Identity, Discrete Math. 178 (1998), 277-281.
[8]	C. D. Savage, A. J. Yee, Euler’s partition theorem and the combinatorics of l-sequences. J. Combin. Theory Ser. A 115 (2008), 967-996.
[9]	X. Xiong and W. J. Keith, Eulers partition theorem for all moduli and new companions to Rogers Ramanujan Andrews-Gordon identities. Ramanujan J. 49 (2019), no. 3 , 555-565.
[10]	Jiang Zeng, The q-variations of Sylvester’s bijection between odd and strict partitions, Ramanujan J. 9 (2005), 289-303.
[11]	K. Alladi, Partitions with non-repeating odd parts and combinatorial identities, Ann. Comb. 20, 1-20 (2016)
[12]	I. Pak, Partition bijections, a survey, Ramanujan J. 12, 5-75 (2006).
[13]	K. Alladi, and A. Berkovich, New weighted Rogers- Ramanujan partition theorems and their implications, Trans. Amer. Math. Soc. 354 (7), 2557-2577 (2002).
[14]	A. J. Yee, On the combinatorics of lecture hall partitions, Ramanujan J. 5, 247-262 (2001).
[15]	A. J. Yee, On the refined lecture hall theorem, Discrete Math. 248, 293-298 (2002).

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APA Style

Ya Gao, Xinhua Xiong. (2022). An Analytic Proof of Some Part of Keith-Xiong’s Theorem. International Journal of Theoretical and Applied Mathematics, 8(1), 14-29. https://doi.org/10.11648/j.ijtam.20220801.12

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ACS Style

Ya Gao; Xinhua Xiong. An Analytic Proof of Some Part of Keith-Xiong’s Theorem. Int. J. Theor. Appl. Math. 2022, 8(1), 14-29. doi: 10.11648/j.ijtam.20220801.12

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AMA Style

Ya Gao, Xinhua Xiong. An Analytic Proof of Some Part of Keith-Xiong’s Theorem. Int J Theor Appl Math. 2022;8(1):14-29. doi: 10.11648/j.ijtam.20220801.12

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@article{10.11648/j.ijtam.20220801.12,
  author = {Ya Gao and Xinhua Xiong},
  title = {An Analytic Proof of Some Part of Keith-Xiong’s Theorem},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {14-29},
  doi = {10.11648/j.ijtam.20220801.12},
  url = {https://doi.org/10.11648/j.ijtam.20220801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220801.12},
  abstract = {In the theory of partitions, Euler’s partition theorem involving odd parts and different parts is one of the famous theorems. It states that the number of partitions of an integer n into odd parts is equal to the number of partitions of n into different parts. By intepretting odd parts as parts congruent to 1 modulo 2, the second author and Keith provided a completely generalization about Euler’s partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan-Andrews-Gordon identities related to this theorem. They gave a combinatorial proof of the theorem by establishing bijection. In this note, we will offer an anclytic view point of this beautiful theorem. We use q-series and generating function theories to provide an analytic style proof for some cases of Keith-Xiong’s theorem. By defining basic units and special units, the basic units in the partitions are divided into two categories, and then the number between the basic units in the special units is classified, and all the cases when m = 3 and alternative sum type (Σ,2) are given, our method is verifying the generating functions of both sides satisfying the same recurrences.},
 year = {2022}
}

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TY  - JOUR
T1  - An Analytic Proof of Some Part of Keith-Xiong’s Theorem
AU  - Ya Gao
AU  - Xinhua Xiong
Y1  - 2022/03/15
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220801.12
DO  - 10.11648/j.ijtam.20220801.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 14
EP  - 29
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220801.12
AB  - In the theory of partitions, Euler’s partition theorem involving odd parts and different parts is one of the famous theorems. It states that the number of partitions of an integer n into odd parts is equal to the number of partitions of n into different parts. By intepretting odd parts as parts congruent to 1 modulo 2, the second author and Keith provided a completely generalization about Euler’s partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan-Andrews-Gordon identities related to this theorem. They gave a combinatorial proof of the theorem by establishing bijection. In this note, we will offer an anclytic view point of this beautiful theorem. We use q-series and generating function theories to provide an analytic style proof for some cases of Keith-Xiong’s theorem. By defining basic units and special units, the basic units in the partitions are divided into two categories, and then the number between the basic units in the special units is classified, and all the cases when m = 3 and alternative sum type (Σ,2) are given, our method is verifying the generating functions of both sides satisfying the same recurrences.
VL  - 8
IS  - 1
ER  -

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Author Information

Ya Gao

College of Science, China Three Gorges University, Yichang, China
Xinhua Xiong

College of Science, China Three Gorges University, Yichang, China

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APA Style

Ya Gao, Xinhua Xiong. (2022). An Analytic Proof of Some Part of Keith-Xiong’s Theorem. International Journal of Theoretical and Applied Mathematics, 8(1), 14-29. https://doi.org/10.11648/j.ijtam.20220801.12

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ACS Style

Ya Gao; Xinhua Xiong. An Analytic Proof of Some Part of Keith-Xiong’s Theorem. Int. J. Theor. Appl. Math. 2022, 8(1), 14-29. doi: 10.11648/j.ijtam.20220801.12

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AMA Style

Ya Gao, Xinhua Xiong. An Analytic Proof of Some Part of Keith-Xiong’s Theorem. Int J Theor Appl Math. 2022;8(1):14-29. doi: 10.11648/j.ijtam.20220801.12

Copy | Download

@article{10.11648/j.ijtam.20220801.12,
  author = {Ya Gao and Xinhua Xiong},
  title = {An Analytic Proof of Some Part of Keith-Xiong’s Theorem},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {8},
  number = {1},
  pages = {14-29},
  doi = {10.11648/j.ijtam.20220801.12},
  url = {https://doi.org/10.11648/j.ijtam.20220801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20220801.12},
  abstract = {In the theory of partitions, Euler’s partition theorem involving odd parts and different parts is one of the famous theorems. It states that the number of partitions of an integer n into odd parts is equal to the number of partitions of n into different parts. By intepretting odd parts as parts congruent to 1 modulo 2, the second author and Keith provided a completely generalization about Euler’s partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan-Andrews-Gordon identities related to this theorem. They gave a combinatorial proof of the theorem by establishing bijection. In this note, we will offer an anclytic view point of this beautiful theorem. We use q-series and generating function theories to provide an analytic style proof for some cases of Keith-Xiong’s theorem. By defining basic units and special units, the basic units in the partitions are divided into two categories, and then the number between the basic units in the special units is classified, and all the cases when m = 3 and alternative sum type (Σ,2) are given, our method is verifying the generating functions of both sides satisfying the same recurrences.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - An Analytic Proof of Some Part of Keith-Xiong’s Theorem
AU  - Ya Gao
AU  - Xinhua Xiong
Y1  - 2022/03/15
PY  - 2022
N1  - https://doi.org/10.11648/j.ijtam.20220801.12
DO  - 10.11648/j.ijtam.20220801.12
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 14
EP  - 29
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20220801.12
AB  - In the theory of partitions, Euler’s partition theorem involving odd parts and different parts is one of the famous theorems. It states that the number of partitions of an integer n into odd parts is equal to the number of partitions of n into different parts. By intepretting odd parts as parts congruent to 1 modulo 2, the second author and Keith provided a completely generalization about Euler’s partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan-Andrews-Gordon identities related to this theorem. They gave a combinatorial proof of the theorem by establishing bijection. In this note, we will offer an anclytic view point of this beautiful theorem. We use q-series and generating function theories to provide an analytic style proof for some cases of Keith-Xiong’s theorem. By defining basic units and special units, the basic units in the partitions are divided into two categories, and then the number between the basic units in the special units is classified, and all the cases when m = 3 and alternative sum type (Σ,2) are given, our method is verifying the generating functions of both sides satisfying the same recurrences.
VL  - 8
IS  - 1
ER  -

Copy | Download