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Research ArticleProof of Some Conjectures of Melham UsingRamanujanrsquos
11205951
Formula
Bipul Kumar Sarmah
Department of Mathematical Sciences Tezpur University Napaam Sonitpur Assam 784028 India
Correspondence should be addressed to Bipul Kumar Sarmah bipultezuernetin
Received 5 February 2014 Accepted 16 June 2014 Published 10 July 2014
Academic Editor Wolfgang zu Castell
Copyright copy 2014 Bipul Kumar SarmahThis is an open access article distributed under theCreativeCommonsAttributionLicensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We employ Ramanujanrsquos11205951formula to prove three conjectures of R S Melham on representation of an integer 119899 as sums of
polygonal numbers
1 Introduction
Jacobirsquos classical two-square theorem is as follows
Theorem 1 (see [1]) Let 119903◻ + ◻(119899) denote the number ofrepresentations of 119899 as a sum of two squares counting orderand sign and let 119889
119894119895(119899) denote the number of positive divisors
of 119899 congruent to 119894modulo 119895 Then
119903 ◻ + ◻ (119899) = 4 (11988914(119899) minus 119889
34(119899)) (1)
The above theorem can also be recasted in terms ofLambert series asinfin
sum
119899=0
119903 ◻ + ◻ (119899) 119902119899= 1 + 4
infin
sum
119899=0
(1199024119899+1
1 minus 1199024119899+1minus1199024119899+3
1 minus 1199024119899+3)
(2)
Similar representation theorems involving squares and tri-angular numbers were found by Dirichlet [2] Lorenz [3]Legendre [4] and Berndt [5] For example the following twotheorems are due to Lorenz and Ramanujan respectively
Theorem 2 (see [3]) Let 119903119897◻ + 119898◻(119899) denote the number ofrepresentations of 119899 as a sum of 119897 times a square and119898 times asquare Then
119903 ◻ + 3◻ (119899) = 2 (11988913(119899) minus 119889
23(119899))
+ 4 (119889412(119899) minus 119889
812(119899))
(3)
Theorem 3 (see [5]) Let 119903119897Δ +119898Δ(119899) denote the number ofrepresentations of 119899 as a sum of 119897 times a triangular numberand119898 times a triangular number Then
Hirschhorn [6 7] obtained forty-five similar identities(including those obtained by Legendre and Ramanujan)involving squares triangular numbers pentagonal numbersand octagonal numbers employing dissection of the 119902-series representations of the identities obtained by JacobiDirichlet and Lorenz In [8] Baruah and the author obtainedtwenty-fivemore such identities involving squares triangularnumbers pentagonal numbers heptagonal numbers octag-onal numbers decagonal numbers hendecagonal numbersdodecagonal numbers and octadecagonal numbers Moreworks on this topic have been done in [9ndash11] In [11] Melhampresented 21 conjectured analogues of Jacobirsquos two-squaretheorem which are verified using computer algorithms In[12] Toh offered a uniform approach to prove these conjec-tures using known formulae for 119903119897◻ + 119898◻(119899) In this paperwe show that some of these conjectures can also be provedby using Ramanujanrsquos famous
11205951formula We prove three
conjectures enlisted in the following theorem which haveappeared as (6) (7) and (8) respectively in [11]
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 738948 6 pageshttpdxdoiorg1011552014738948
2 International Journal of Mathematics and Mathematical Sciences
Theorem 4 Considerinfin
sum
119899=0
119903 Δ + 5Δ (119899) 119902119899=
infin
sum
119899=0
(1199023119899+ 1199027119899+1
1 minus 11990220119899+5minus11990213119899+9+ 11990217119899+12
1 minus 11990220119899+15
)
(5)infin
sum
119899=0
119903 Δ + 6Δ (119899) 119902119899=
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(6)infin
sum
119899=0
119903 2Δ + 3Δ (119899) 119902119899=
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(7)
The next section of this paper is devoted to notationsdefinitions and preliminary results
2 Notations and Preliminary Results
Ramanujanrsquos general theta-function 119891(119886 119887) is defined by [5page 34 (181)]
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and 119887 = 11990225in (15) and proceeding as in case of (22) we find thatinfin
Proof of Conjecture (6) In view of (17) we rewrite (6) as
120595 (119902) 120595 (1199026) =
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(32)
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and 119887 = 11990225in (15) and proceeding as in case of (22) we find thatinfin
Proof of Conjecture (6) In view of (17) we rewrite (6) as
120595 (119902) 120595 (1199026) =
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(32)
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
Replacing 119902 by 11990220 and then setting 119911 = 1199023 119886 = 1199025 and 119887 = 11990225in (15) and proceeding as in case of (22) we find thatinfin
Proof of Conjecture (6) In view of (17) we rewrite (6) as
120595 (119902) 120595 (1199026) =
infin
sum
119899=0
(1199027119899
1 minus 11990224119899+3+1199025119899+1
1 minus 11990224119899+9
minus11990219119899+11
1 minus 11990224119899+15minus11990217119899+14
1 minus 11990224119899+21)
(32)
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
4 International Journal of Mathematics and Mathematical Sciences
Replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199023 and 119887 = 11990227in (15) and proceeding as in case of (22) we find that
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199029 and119887 = 11990233 in (15) and proceeding in a way similar to obtaining
Proof of Conjecture (7) In view of (17) we rewrite (7) as
120595 (1199022) 120595 (119902
3) =
infin
sum
119899=0
(1199025119899
1 minus 11990224119899+3+1199027119899+2
1 minus 11990224119899+9
minus11990217119899+10
1 minus 11990224119899+15minus11990219119899+16
1 minus 11990224119899+21)
(40)
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
International Journal of Mathematics and Mathematical Sciences 5
Replacing 119902 by 11990224 and then setting 119911 = 1199025 119886 = 1199023 and119887 = 11990227 in (15) and proceeding as in case of (22) we find that
Also replacing 119902 by 11990224 and then setting 119911 = 1199027 119886 = 1199029and 119887 = 11990233 in (15) and proceeding similar to (22) we obtain
All conjectures in [11] can easily be reformulated as thetafunction identity using Ramanujanrsquos
11205951formula However
these identities might be too complicated to actually have aproof
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C G J Jacobi Fundamenta Nova Theoriae Functionum Ellipti-carum vol 107 of Werke I 162-163 Letter to Legendre 991828Werke I 424 1829
[2] P G L Dirichlet J Math 21 (1840) 3 6 Werke 463 466[3] L Lorenz ldquoBidrag til tallenes theorirdquo Tidsskrift for Mathematik
vol 3 no 1 pp 97ndash114 1871[4] A M Legendre Traite des Fonctions Elliptiques et des Integrales
Euleriennes III Huzard-Courcier Paris France 1828[5] B C Berndt Ramanujanrsquos Notebooks Part III Springer New
York NY USA 1991
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013
6 International Journal of Mathematics and Mathematical Sciences
[6] M D Hirschhorn ldquoThe number of representations of a numberby various formsrdquo Discrete Mathematics vol 298 no 1ndash3 pp205ndash211 2005
[7] M D Hirschhorn ldquoThe number of representation of a numberby various forms involving triangles squares pentagons andoctagonsrdquo in Ramanujan Rediscovered N D Baruah B CBerndt S Cooper T Huber and M Schlosser Eds RMS Lec-ture Note Series No 14 pp 113ndash124 Ramanujan MathematicalSociety 2010
[8] ND Baruah and B K Sarmah ldquoThenumber of representationsof a number as sums of various polygonal numbersrdquo Integersvol 12 2012
[9] H Y Lam ldquoThe number of representations by sums of squaresand triangular numbersrdquo Integers vol 7 article A28 2007
[10] R S Melham ldquoAnalogues of two classical theorems on therepresentations of a numberrdquo Integers vol 8 article A51 2008
[11] R S Melham ldquoAnalogues of Jacobirsquos two-square theorem aninformal accountrdquo Integers vol 10 no 1 pp 83ndash100 2010
[12] P C Toh ldquoOn representations by figurate numbers a uniformapproach to the conjectures of Melhamrdquo International Journalof Number Theory vol 9 no 4 pp 1055ndash1071 2013