Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences
In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn＋1/zn}is increasing（resp.,decreasing）,then the sequence {n√zn} is strictly increasing（resp.,decreasing） s...
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| Published in | Science China. Mathematics Vol. 57; no. 11; pp. 2429 - 2435 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Heidelberg
Science China Press
01.11.2014
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| Online Access | Get full text |
| ISSN | 1674-7283 1869-1862 |
| DOI | 10.1007/s11425-014-4851-x |
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| Abstract | In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn＋1/zn}is increasing（resp.,decreasing）,then the sequence {n√zn} is strictly increasing（resp.,decreasing） subject to a certain initial condition. We also give some sufficient conditions when {zn＋1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach. |
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| AbstractList | In 2012, Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form (ProQuest: Formulae and/or non-USASCII text omitted), where {z sub(n)} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {z sub(n)+1/z sub(n)} is increasing (resp., decreasing), then the sequence (ProQuest: Formulae and/or non-USASCII text omitted) is strictly increasing (resp., decreasing) subject to a certain initial condition. We also give some sufficient conditions when {z sub(n)+1/z sub(n)} is increasing, which is equivalent to the log-convexity of {z sub(n)}. As consequences, a series of conjectures of Zhi-Wei Sun are verified in a unified approach. In 2012, Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form , where { z n } is a familiar number-theoretic or combinatorial sequence. We show that if the sequence { z n +1 / z n } is increasing (resp., decreasing), then the sequence is strictly increasing (resp., decreasing) subject to a certain initial condition. We also give some sufficient conditions when { z n +1 / z n } is increasing, which is equivalent to the log-convexity of { z n }. As consequences, a series of conjectures of Zhi-Wei Sun are verified in a unified approach. In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn＋1/zn}is increasing（resp.,decreasing）,then the sequence {n√zn} is strictly increasing（resp.,decreasing） subject to a certain initial condition. We also give some sufficient conditions when {zn＋1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach. |
| Author | WANG Yi ZHU BaoXuan |
| AuthorAffiliation | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China School of Mathematical Sciences, Jiangsu Normal University, Xuzhou 221116, China |
| Author_xml | – sequence: 1 givenname: Yi surname: Wang fullname: Wang, Yi email: wangyi@dlut.edu.cn organization: School of Mathematical Sciences, Dalian University of Technology – sequence: 2 givenname: BaoXuan surname: Zhu fullname: Zhu, BaoXuan organization: School of Mathematical Sciences, Jiangsu Normal University |
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| Notes | sequences; monotonicity; log-convexity; log-concavity In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn＋1/zn}is increasing（resp.,decreasing）,then the sequence {n√zn} is strictly increasing（resp.,decreasing） subject to a certain initial condition. We also give some sufficient conditions when {zn＋1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach. 11-1787/N ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
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| References | DavenportHPólyaGOn the product of two power serieCanad J Math194911510.4153/CJM-1949-001-10037.32505 JanoskiJ EA collection of problems in combinatorics2012South CarolinaClemson University EngelKOn the average rank of an element in a filter of the partition latticeJ Combin Theory Ser A1994656778125526410.1016/0097-3165(94)90038-80795.05051 Amdeberhan T, Moll V H, Vignat C. A probabilistic interpretation of a sequence related to Narayana polynomials. Online J Anal Comb, 2013, 8: 25pp DošlićTSeven (lattice) paths to log-convexityActa Appl Math201011013731392263917610.1007/s10440-009-9515-41225.05018 LassalleMTwo integer sequences related to Catalan numbersJ Combin Theory Ser A2012119923935288123510.1016/j.jcta.2012.01.0021241.05003 Chen W Y C, Guo J J F, Wang L X W. Zeta functions and the log-behavior of combinatorial sequences. Proc Edinburgh Math Soc (2), in press ChenW Y CGuoJ J FWangL X WInfinitely logarithmically monotonic combinatorial sequencesAdv Appl Math20145299120313192810.1016/j.aam.2013.08.0031281.05019 SunZ-WKanemitsuSLiHLiuJConjectures involving arithmetical sequencesNumbers Theory: Arithmetic in Shangri-La, Proceedings 6th China-Japan Seminar2013SingaporeWorld Scientific244258 WangYYehY NPolynomials with real zeros and Pólya frequency sequencesJ Combin Theory Ser A20051096374211019810.1016/j.jcta.2004.07.0081057.05007 ChenW Y CRecent developments of log-concavity and q-log-concavity of combinatorial polynomials2010San Francisco, CAA talk given at the 22nd Inter Confer on Formal Power Series and Algebraic Combin LiuL LWangYOn the log-convexity of combinatorial sequencesAdv Appl Math20073945347610.1016/j.aam.2006.11.0021131.05010 GrahamR LKnuthD EPatashnikOConcrete Mathematics: A Foundation for Computer Science19942nd ed.Reading, MassachusettsAddison-Wesley0836.00001 Hou Q H, Sun Z-W, Wen H M. On monotonicity of some combinatorial sequences. Publ Math Debrecen, in press WangYYehY NLog-concavity and LC-positivityJ Combin Theory Ser A2007114195210229308710.1016/j.jcta.2006.02.0011109.11019 LiuL LWangYA unified approach to polynomial sequences with only real zerosAdv Appl Math20073854256010.1016/j.aam.2006.02.0031123.05009 BrentiFLog-concave and unimodal sequences in algebra, combinatorics, and geometry: An updateContemp Math19941787189131057510.1090/conm/178/01893 StanleyR PLog-concave and unimodal sequences in algebra, combinatorics, and geometryAnn New York Acad Sci198957650053410.1111/j.1749-6632.1989.tb16434.x SunZ-WOn a sequence involving sums of primesBull Aust Math Soc201388197205310970910.1017/S000497271200098606234290 ChenW Y CTangR LWangL X WThe q-log-convexity of Narayana polynomials of type BAdv Appl Math20104485110257684110.1016/j.aam.2009.03.0041230.05276 ZhuB XLog-convexity and strong q-log-convexity for some triangular arraysAdv Appl Math20135059560610.1016/j.aam.2012.11.0031277.05014 Sloane N J A. The On-Line Encyclopedia of Integer Sequences. http://oeis.org LucaFStănicăPOn some conjectures on the monotonicity of some arithematical sequencesJ Combin Number Theory2012411512306242441 J E Janoski (4851_CR12) 2012 K Engel (4851_CR9) 1994; 65 4851_CR11 M Lassalle (4851_CR13) 2012; 119 L L Liu (4851_CR14) 2007; 38 4851_CR1 F Luca (4851_CR16) 2012; 4 F Brenti (4851_CR2) 1994; 178 W Y C Chen (4851_CR5) 2014; 52 R P Stanley (4851_CR18) 1989; 576 H Davenport (4851_CR7) 1949; 1 4851_CR4 Y Wang (4851_CR21) 2005; 109 W Y C Chen (4851_CR6) 2010; 44 R L Graham (4851_CR10) 1994 T Došlić (4851_CR8) 2010; 110 B X Zhu (4851_CR23) 2013; 50 Z-W Sun (4851_CR19) 2013; 88 Z-W Sun (4851_CR20) 2013 L L Liu (4851_CR15) 2007; 39 Y Wang (4851_CR22) 2007; 114 W Y C Chen (4851_CR3) 2010 4851_CR17 |
| References_xml | – reference: SunZ-WKanemitsuSLiHLiuJConjectures involving arithmetical sequencesNumbers Theory: Arithmetic in Shangri-La, Proceedings 6th China-Japan Seminar2013SingaporeWorld Scientific244258 – reference: Chen W Y C, Guo J J F, Wang L X W. Zeta functions and the log-behavior of combinatorial sequences. Proc Edinburgh Math Soc (2), in press – reference: Sloane N J A. The On-Line Encyclopedia of Integer Sequences. http://oeis.org/ – reference: EngelKOn the average rank of an element in a filter of the partition latticeJ Combin Theory Ser A1994656778125526410.1016/0097-3165(94)90038-80795.05051 – reference: WangYYehY NPolynomials with real zeros and Pólya frequency sequencesJ Combin Theory Ser A20051096374211019810.1016/j.jcta.2004.07.0081057.05007 – reference: ChenW Y CRecent developments of log-concavity and q-log-concavity of combinatorial polynomials2010San Francisco, CAA talk given at the 22nd Inter Confer on Formal Power Series and Algebraic Combin – reference: WangYYehY NLog-concavity and LC-positivityJ Combin Theory Ser A2007114195210229308710.1016/j.jcta.2006.02.0011109.11019 – reference: LassalleMTwo integer sequences related to Catalan numbersJ Combin Theory Ser A2012119923935288123510.1016/j.jcta.2012.01.0021241.05003 – reference: LiuL LWangYA unified approach to polynomial sequences with only real zerosAdv Appl Math20073854256010.1016/j.aam.2006.02.0031123.05009 – reference: BrentiFLog-concave and unimodal sequences in algebra, combinatorics, and geometry: An updateContemp Math19941787189131057510.1090/conm/178/01893 – reference: JanoskiJ EA collection of problems in combinatorics2012South CarolinaClemson University – reference: ChenW Y CGuoJ J FWangL X WInfinitely logarithmically monotonic combinatorial sequencesAdv Appl Math20145299120313192810.1016/j.aam.2013.08.0031281.05019 – reference: DavenportHPólyaGOn the product of two power serieCanad J Math194911510.4153/CJM-1949-001-10037.32505 – reference: SunZ-WOn a sequence involving sums of primesBull Aust Math Soc201388197205310970910.1017/S000497271200098606234290 – reference: GrahamR LKnuthD EPatashnikOConcrete Mathematics: A Foundation for Computer Science19942nd ed.Reading, MassachusettsAddison-Wesley0836.00001 – reference: Hou Q H, Sun Z-W, Wen H M. On monotonicity of some combinatorial sequences. Publ Math Debrecen, in press – reference: LucaFStănicăPOn some conjectures on the monotonicity of some arithematical sequencesJ Combin Number Theory2012411512306242441 – reference: DošlićTSeven (lattice) paths to log-convexityActa Appl Math201011013731392263917610.1007/s10440-009-9515-41225.05018 – reference: LiuL LWangYOn the log-convexity of combinatorial sequencesAdv Appl Math20073945347610.1016/j.aam.2006.11.0021131.05010 – reference: ChenW Y CTangR LWangL X WThe q-log-convexity of Narayana polynomials of type BAdv Appl Math20104485110257684110.1016/j.aam.2009.03.0041230.05276 – reference: StanleyR PLog-concave and unimodal sequences in algebra, combinatorics, and geometryAnn New York Acad Sci198957650053410.1111/j.1749-6632.1989.tb16434.x – reference: Amdeberhan T, Moll V H, Vignat C. A probabilistic interpretation of a sequence related to Narayana polynomials. Online J Anal Comb, 2013, 8: 25pp – reference: ZhuB XLog-convexity and strong q-log-convexity for some triangular arraysAdv Appl Math20135059560610.1016/j.aam.2012.11.0031277.05014 – volume: 110 start-page: 1373 year: 2010 ident: 4851_CR8 publication-title: Acta Appl Math doi: 10.1007/s10440-009-9515-4 – volume-title: Concrete Mathematics: A Foundation for Computer Science year: 1994 ident: 4851_CR10 – volume: 88 start-page: 197 year: 2013 ident: 4851_CR19 publication-title: Bull Aust Math Soc doi: 10.1017/S0004972712000986 – volume-title: A collection of problems in combinatorics year: 2012 ident: 4851_CR12 – volume: 38 start-page: 542 year: 2007 ident: 4851_CR14 publication-title: Adv Appl Math doi: 10.1016/j.aam.2006.02.003 – volume: 119 start-page: 923 year: 2012 ident: 4851_CR13 publication-title: J Combin Theory Ser A doi: 10.1016/j.jcta.2012.01.002 – volume: 52 start-page: 99 year: 2014 ident: 4851_CR5 publication-title: Adv Appl Math doi: 10.1016/j.aam.2013.08.003 – volume: 1 start-page: 1 year: 1949 ident: 4851_CR7 publication-title: Canad J Math doi: 10.4153/CJM-1949-001-1 – volume: 109 start-page: 63 year: 2005 ident: 4851_CR21 publication-title: J Combin Theory Ser A doi: 10.1016/j.jcta.2004.07.008 – volume: 44 start-page: 85 year: 2010 ident: 4851_CR6 publication-title: Adv Appl Math doi: 10.1016/j.aam.2009.03.004 – volume: 4 start-page: 115 year: 2012 ident: 4851_CR16 publication-title: J Combin Number Theory – volume: 178 start-page: 71 year: 1994 ident: 4851_CR2 publication-title: Contemp Math doi: 10.1090/conm/178/01893 – volume: 39 start-page: 453 year: 2007 ident: 4851_CR15 publication-title: Adv Appl Math doi: 10.1016/j.aam.2006.11.002 – volume-title: Recent developments of log-concavity and q-log-concavity of combinatorial polynomials year: 2010 ident: 4851_CR3 – ident: 4851_CR11 – volume: 50 start-page: 595 year: 2013 ident: 4851_CR23 publication-title: Adv Appl Math doi: 10.1016/j.aam.2012.11.003 – ident: 4851_CR4 – volume: 576 start-page: 500 year: 1989 ident: 4851_CR18 publication-title: Ann New York Acad Sci doi: 10.1111/j.1749-6632.1989.tb16434.x – start-page: 244 volume-title: Numbers Theory: Arithmetic in Shangri-La, Proceedings 6th China-Japan Seminar year: 2013 ident: 4851_CR20 doi: 10.1142/9789814452458_0014 – volume: 65 start-page: 67 year: 1994 ident: 4851_CR9 publication-title: J Combin Theory Ser A doi: 10.1016/0097-3165(94)90038-8 – ident: 4851_CR1 – volume: 114 start-page: 195 year: 2007 ident: 4851_CR22 publication-title: J Combin Theory Ser A doi: 10.1016/j.jcta.2006.02.001 – ident: 4851_CR17 |
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