Relations among representations of integers by certain quadratic forms

Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Baruah, Nayandeep Deka [verfasserIn] Das, Hirakjyoti

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2021

Schlagwörter:	Theta function Cubic theta function Quadratic form

Anmerkung:	© The Indian National Science Academy 2021

Übergeordnetes Werk:	Enthalten in: Indian journal of pure and applied mathematics - New Delhi : Acad., 1970, 53(2021), 3 vom: 05. Aug., Seite 672-682
Übergeordnetes Werk:	volume:53 ; year:2021 ; number:3 ; day:05 ; month:08 ; pages:672-682

Links:	Volltext

DOI / URN:	10.1007/s13226-021-00158-w

Katalog-ID:	SPR050955012

Internformat


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100	1		\|a Baruah, Nayandeep Deka \|e verfasserin \|4 aut
245	1	0	\|a Relations among representations of integers by certain quadratic forms
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520			\|a Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$
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700	1		\|a Das, Hirakjyoti \|0 (orcid)0000-0002-5540-5625 \|4 aut
773	0	8	\|i Enthalten in \|t Indian journal of pure and applied mathematics \|d New Delhi : Acad., 1970 \|g 53(2021), 3 vom: 05. Aug., Seite 672-682 \|w (DE-627)558697410 \|w (DE-600)2410736-0 \|x 0975-7465 \|7 nnns
773	1	8	\|g volume:53 \|g year:2021 \|g number:3 \|g day:05 \|g month:08 \|g pages:672-682
856	4	0	\|u https://dx.doi.org/10.1007/s13226-021-00158-w \|z lizenzpflichtig \|3 Volltext
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912			\|a GBV_ILN_22
912			\|a GBV_ILN_23
912			\|a GBV_ILN_24
912			\|a GBV_ILN_31
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_65
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_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_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_4012
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4126
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4328
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
912			\|a GBV_ILN_4367
912			\|a GBV_ILN_4393
912			\|a GBV_ILN_4700
951			\|a AR
952			\|d 53 \|j 2021 \|e 3 \|b 05 \|c 08 \|h 672-682

Indexfelder

author_variant	n d b nd ndb h d hd
matchkey_str	article:09757465:2021----::eainaogersnainoitgrbcr
hierarchy_sort_str	2021
publishDate	2021
allfields	10.1007/s13226-021-00158-w doi (DE-627)SPR050955012 (SPR)s13226-021-00158-w-e DE-627 ger DE-627 rakwb eng Baruah, Nayandeep Deka verfasserin aut Relations among representations of integers by certain quadratic forms 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ Theta function (dpeaa)DE-He213 Cubic theta function (dpeaa)DE-He213 Quadratic form (dpeaa)DE-He213 Das, Hirakjyoti (orcid)0000-0002-5540-5625 aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 53(2021), 3 vom: 05. Aug., Seite 672-682 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:53 year:2021 number:3 day:05 month:08 pages:672-682 https://dx.doi.org/10.1007/s13226-021-00158-w 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_65 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_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_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_4012 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_4367 GBV_ILN_4393 GBV_ILN_4700 AR 53 2021 3 05 08 672-682
spelling	10.1007/s13226-021-00158-w doi (DE-627)SPR050955012 (SPR)s13226-021-00158-w-e DE-627 ger DE-627 rakwb eng Baruah, Nayandeep Deka verfasserin aut Relations among representations of integers by certain quadratic forms 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ Theta function (dpeaa)DE-He213 Cubic theta function (dpeaa)DE-He213 Quadratic form (dpeaa)DE-He213 Das, Hirakjyoti (orcid)0000-0002-5540-5625 aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 53(2021), 3 vom: 05. Aug., Seite 672-682 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:53 year:2021 number:3 day:05 month:08 pages:672-682 https://dx.doi.org/10.1007/s13226-021-00158-w 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_65 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_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_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_4012 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_4367 GBV_ILN_4393 GBV_ILN_4700 AR 53 2021 3 05 08 672-682
allfields_unstemmed	10.1007/s13226-021-00158-w doi (DE-627)SPR050955012 (SPR)s13226-021-00158-w-e DE-627 ger DE-627 rakwb eng Baruah, Nayandeep Deka verfasserin aut Relations among representations of integers by certain quadratic forms 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ Theta function (dpeaa)DE-He213 Cubic theta function (dpeaa)DE-He213 Quadratic form (dpeaa)DE-He213 Das, Hirakjyoti (orcid)0000-0002-5540-5625 aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 53(2021), 3 vom: 05. Aug., Seite 672-682 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:53 year:2021 number:3 day:05 month:08 pages:672-682 https://dx.doi.org/10.1007/s13226-021-00158-w 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_65 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_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_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_4012 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_4367 GBV_ILN_4393 GBV_ILN_4700 AR 53 2021 3 05 08 672-682
allfieldsGer	10.1007/s13226-021-00158-w doi (DE-627)SPR050955012 (SPR)s13226-021-00158-w-e DE-627 ger DE-627 rakwb eng Baruah, Nayandeep Deka verfasserin aut Relations among representations of integers by certain quadratic forms 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ Theta function (dpeaa)DE-He213 Cubic theta function (dpeaa)DE-He213 Quadratic form (dpeaa)DE-He213 Das, Hirakjyoti (orcid)0000-0002-5540-5625 aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 53(2021), 3 vom: 05. Aug., Seite 672-682 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:53 year:2021 number:3 day:05 month:08 pages:672-682 https://dx.doi.org/10.1007/s13226-021-00158-w 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_65 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_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_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_4012 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_4367 GBV_ILN_4393 GBV_ILN_4700 AR 53 2021 3 05 08 672-682
allfieldsSound	10.1007/s13226-021-00158-w doi (DE-627)SPR050955012 (SPR)s13226-021-00158-w-e DE-627 ger DE-627 rakwb eng Baruah, Nayandeep Deka verfasserin aut Relations among representations of integers by certain quadratic forms 2021 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Indian National Science Academy 2021 Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ Theta function (dpeaa)DE-He213 Cubic theta function (dpeaa)DE-He213 Quadratic form (dpeaa)DE-He213 Das, Hirakjyoti (orcid)0000-0002-5540-5625 aut Enthalten in Indian journal of pure and applied mathematics New Delhi : Acad., 1970 53(2021), 3 vom: 05. Aug., Seite 672-682 (DE-627)558697410 (DE-600)2410736-0 0975-7465 nnns volume:53 year:2021 number:3 day:05 month:08 pages:672-682 https://dx.doi.org/10.1007/s13226-021-00158-w 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_65 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_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_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_4012 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_4367 GBV_ILN_4393 GBV_ILN_4700 AR 53 2021 3 05 08 672-682
language	English
source	Enthalten in Indian journal of pure and applied mathematics 53(2021), 3 vom: 05. Aug., Seite 672-682 volume:53 year:2021 number:3 day:05 month:08 pages:672-682
sourceStr	Enthalten in Indian journal of pure and applied mathematics 53(2021), 3 vom: 05. Aug., Seite 672-682 volume:53 year:2021 number:3 day:05 month:08 pages:672-682
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Theta function Cubic theta function Quadratic form
isfreeaccess_bool	false
container_title	Indian journal of pure and applied mathematics
authorswithroles_txt_mv	Baruah, Nayandeep Deka @@aut@@ Das, Hirakjyoti @@aut@@
publishDateDaySort_date	2021-08-05T00:00:00Z
hierarchy_top_id	558697410
id	SPR050955012
language_de	englisch
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Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. 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author	Baruah, Nayandeep Deka
spellingShingle	Baruah, Nayandeep Deka misc Theta function misc Cubic theta function misc Quadratic form Relations among representations of integers by certain quadratic forms
authorStr	Baruah, Nayandeep Deka
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illustrated	Not Illustrated
issn	0975-7465
topic_title	Relations among representations of integers by certain quadratic forms Theta function (dpeaa)DE-He213 Cubic theta function (dpeaa)DE-He213 Quadratic form (dpeaa)DE-He213
topic	misc Theta function misc Cubic theta function misc Quadratic form
topic_unstemmed	misc Theta function misc Cubic theta function misc Quadratic form
topic_browse	misc Theta function misc Cubic theta function misc Quadratic form
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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hierarchy_parent_title	Indian journal of pure and applied mathematics
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title	Relations among representations of integers by certain quadratic forms
ctrlnum	(DE-627)SPR050955012 (SPR)s13226-021-00158-w-e
title_full	Relations among representations of integers by certain quadratic forms
author_sort	Baruah, Nayandeep Deka
journal	Indian journal of pure and applied mathematics
journalStr	Indian journal of pure and applied mathematics
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author_browse	Baruah, Nayandeep Deka Das, Hirakjyoti
container_volume	53
format_se	Elektronische Aufsätze
author-letter	Baruah, Nayandeep Deka
doi_str_mv	10.1007/s13226-021-00158-w
normlink	(ORCID)0000-0002-5540-5625
normlink_prefix_str_mv	(orcid)0000-0002-5540-5625
title_sort	relations among representations of integers by certain quadratic forms
title_auth	Relations among representations of integers by certain quadratic forms
abstract	Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ © The Indian National Science Academy 2021
abstractGer	Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ © The Indian National Science Academy 2021
abstract_unstemmed	Abstract Let %$N(c_1,c_2,\ldots , c_k;n),%%%$r(c_1,c_2,\ldots , c_k;n),%$ and %$t(c_1,c_2,\ldots , c_k;n)%$ count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,%$\begin{aligned}&c_1\left( x_1^2+x_1 x_2+x_2^2\right) +c_2\left( x_3^2+x_3 x_4+x_4^2\right) +\cdots +c_k\big (x_{2k-1}^2+x_{2k-1}x_{2k}+x_{2k}^2\big ), \\&\quad c_1x_1^2+c_2x_2^2+\cdots +c_kx_k^2, \\&\quad c_1x_1(x_1+1)/2+c_2x_2(x_2+1)/2+\cdots +c_kx_k(x_k+1)/2, \end{aligned}%$respectively, where %$c_i%$’s and %$x_i%$’s are integers. Using Ramanujan’s theta functions %$\varphi (q)%$, %$\psi (q)%$, Borweins’ cubic theta function a(q), and simple dissecting techniques, we find relations satisfied by %$N(c_1,c_2,\ldots , c_k;n)%$, %$r(c_1,c_2,\ldots , c_{2k};n)%$, and %$t(c_1,c_2,\ldots , c_{2k};n)%$ when %$1<2k \le 8%$. We also deduce some new explicit formulas for %$r(c_1,c_2,c_3,c_4;n)%$ for some particular n when %$c_i\in \{1,2,3,6,8,9,16,18,24,48\}.%$ © The Indian National Science Academy 2021
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container_issue	3
title_short	Relations among representations of integers by certain quadratic forms
url	https://dx.doi.org/10.1007/s13226-021-00158-w
remote_bool	true
author2	Das, Hirakjyoti
author2Str	Das, Hirakjyoti
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doi_str	10.1007/s13226-021-00158-w
up_date	2024-07-03T18:51:30.452Z
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Relations among representations of integers by certain quadratic forms

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