On the concavity properties of certain arithmetic sequences and polynomials

Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alph...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Zhu, Bao-Xuan [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2023

Schlagwörter:	Log-concavity 3- -Log-concavity 3-Log-concavity Total positivity

Anmerkung:	© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Übergeordnetes Werk:	Enthalten in: Mathematische Zeitschrift - Berlin : Springer, 1918, 305(2023), 3 vom: 04. Okt.
Übergeordnetes Werk:	volume:305 ; year:2023 ; number:3 ; day:04 ; month:10

Links:	Volltext

DOI / URN:	10.1007/s00209-023-03361-z

Katalog-ID:	SPR053300289

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245	1	0	\|a On the concavity properties of certain arithmetic sequences and polynomials
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520			\|a Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question.
650		4	\|a Log-concavity \|7 (dpeaa)DE-He213
650		4	\|a 3- \|7 (dpeaa)DE-He213
650		4	\|a -Log-concavity \|7 (dpeaa)DE-He213
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650		4	\|a Total positivity \|7 (dpeaa)DE-He213
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912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_267
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2018
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
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912			\|a GBV_ILN_4125
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912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
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912			\|a GBV_ILN_4307
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912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
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912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
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allfields	10.1007/s00209-023-03361-z doi (DE-627)SPR053300289 (SPR)s00209-023-03361-z-e DE-627 ger DE-627 rakwb eng Zhu, Bao-Xuan verfasserin aut On the concavity properties of certain arithmetic sequences and polynomials 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. Log-concavity (dpeaa)DE-He213 3- (dpeaa)DE-He213 -Log-concavity (dpeaa)DE-He213 3-Log-concavity (dpeaa)DE-He213 Total positivity (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 305(2023), 3 vom: 04. Okt. (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:305 year:2023 number:3 day:04 month:10 https://dx.doi.org/10.1007/s00209-023-03361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 305 2023 3 04 10
spelling	10.1007/s00209-023-03361-z doi (DE-627)SPR053300289 (SPR)s00209-023-03361-z-e DE-627 ger DE-627 rakwb eng Zhu, Bao-Xuan verfasserin aut On the concavity properties of certain arithmetic sequences and polynomials 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. Log-concavity (dpeaa)DE-He213 3- (dpeaa)DE-He213 -Log-concavity (dpeaa)DE-He213 3-Log-concavity (dpeaa)DE-He213 Total positivity (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 305(2023), 3 vom: 04. Okt. (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:305 year:2023 number:3 day:04 month:10 https://dx.doi.org/10.1007/s00209-023-03361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 305 2023 3 04 10
allfields_unstemmed	10.1007/s00209-023-03361-z doi (DE-627)SPR053300289 (SPR)s00209-023-03361-z-e DE-627 ger DE-627 rakwb eng Zhu, Bao-Xuan verfasserin aut On the concavity properties of certain arithmetic sequences and polynomials 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. Log-concavity (dpeaa)DE-He213 3- (dpeaa)DE-He213 -Log-concavity (dpeaa)DE-He213 3-Log-concavity (dpeaa)DE-He213 Total positivity (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 305(2023), 3 vom: 04. Okt. (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:305 year:2023 number:3 day:04 month:10 https://dx.doi.org/10.1007/s00209-023-03361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 305 2023 3 04 10
allfieldsGer	10.1007/s00209-023-03361-z doi (DE-627)SPR053300289 (SPR)s00209-023-03361-z-e DE-627 ger DE-627 rakwb eng Zhu, Bao-Xuan verfasserin aut On the concavity properties of certain arithmetic sequences and polynomials 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. Log-concavity (dpeaa)DE-He213 3- (dpeaa)DE-He213 -Log-concavity (dpeaa)DE-He213 3-Log-concavity (dpeaa)DE-He213 Total positivity (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 305(2023), 3 vom: 04. Okt. (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:305 year:2023 number:3 day:04 month:10 https://dx.doi.org/10.1007/s00209-023-03361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 305 2023 3 04 10
allfieldsSound	10.1007/s00209-023-03361-z doi (DE-627)SPR053300289 (SPR)s00209-023-03361-z-e DE-627 ger DE-627 rakwb eng Zhu, Bao-Xuan verfasserin aut On the concavity properties of certain arithmetic sequences and polynomials 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. Log-concavity (dpeaa)DE-He213 3- (dpeaa)DE-He213 -Log-concavity (dpeaa)DE-He213 3-Log-concavity (dpeaa)DE-He213 Total positivity (dpeaa)DE-He213 Enthalten in Mathematische Zeitschrift Berlin : Springer, 1918 305(2023), 3 vom: 04. Okt. (DE-627)254630812 (DE-600)1462134-4 1432-1823 nnns volume:305 year:2023 number:3 day:04 month:10 https://dx.doi.org/10.1007/s00209-023-03361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 305 2023 3 04 10
language	English
source	Enthalten in Mathematische Zeitschrift 305(2023), 3 vom: 04. Okt. volume:305 year:2023 number:3 day:04 month:10
sourceStr	Enthalten in Mathematische Zeitschrift 305(2023), 3 vom: 04. Okt. volume:305 year:2023 number:3 day:04 month:10
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Log-concavity 3- -Log-concavity 3-Log-concavity Total positivity
isfreeaccess_bool	false
container_title	Mathematische Zeitschrift
authorswithroles_txt_mv	Zhu, Bao-Xuan @@aut@@
publishDateDaySort_date	2023-10-04T00:00:00Z
hierarchy_top_id	254630812
id	SPR053300289
language_de	englisch
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author	Zhu, Bao-Xuan
spellingShingle	Zhu, Bao-Xuan misc Log-concavity misc 3- misc -Log-concavity misc 3-Log-concavity misc Total positivity On the concavity properties of certain arithmetic sequences and polynomials
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topic_title	On the concavity properties of certain arithmetic sequences and polynomials Log-concavity (dpeaa)DE-He213 3- (dpeaa)DE-He213 -Log-concavity (dpeaa)DE-He213 3-Log-concavity (dpeaa)DE-He213 Total positivity (dpeaa)DE-He213
topic	misc Log-concavity misc 3- misc -Log-concavity misc 3-Log-concavity misc Total positivity
topic_unstemmed	misc Log-concavity misc 3- misc -Log-concavity misc 3-Log-concavity misc Total positivity
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title	On the concavity properties of certain arithmetic sequences and polynomials
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title_full	On the concavity properties of certain arithmetic sequences and polynomials
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title_sort	on the concavity properties of certain arithmetic sequences and polynomials
title_auth	On the concavity properties of certain arithmetic sequences and polynomials
abstract	Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstractGer	Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
abstract_unstemmed	Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. Finally, we pose an interesting question. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
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title_short	On the concavity properties of certain arithmetic sequences and polynomials
url	https://dx.doi.org/10.1007/s00209-023-03361-z
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up_date	2024-07-03T18:31:50.623Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR053300289</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20231028064701.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">231006s2023 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00209-023-03361-z</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR053300289</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00209-023-03361-z-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Zhu, Bao-Xuan</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">On the concavity properties of certain arithmetic sequences and polynomials</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2023</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Given a sequence %$\alpha =(a_k)_{k\ge 0}%$ of nonnegative numbers, define a new sequence %${\mathcal {L}}(\alpha )=(b_k)_{k\ge 0}%$ by %$b_{k}=a^2_{k}-a_{k-1}a_{k+1}%$. The sequence %$\alpha %$ is called r-log-concave if %${\mathcal {L}}^{i}(\alpha )=\mathcal {L}({\mathcal {L}}^{i-1}(\alpha ))%$ is a nonnegative sequence for all %$1\le i\le r%$. In this paper, we study the r-log-concavity and its q-analogue for %$r=2,3%$ using total positivity of matrices. We show the 6-log-concavity of the Taylor coefficients of the Riemann %$\xi %$-function. We give some criteria for r-q-log-concavity for %$r=2,3%$. As applications, we get 3-q-log-concavity of q-binomial coefficients and different q-Stirling numbers of two kinds, which extends results for q-log-concavity. We also present some results for r-q-log-concavity from the linear transformations. 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