{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T00:09:15Z","timestamp":1767139755101,"version":"build-2238731810"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"21","license":[{"start":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T00:00:00Z","timestamp":1682380800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T00:00:00Z","timestamp":1682380800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Manipal Academy of Higher Education, Manipal"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2023,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In 2009, M. S. M. Naika et al. obtained Schl\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\ddot{a}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mover>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>\u00a8<\/mml:mo>\n                          <\/mml:mover>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    fli type modular equations of degree 4. Motivated by their work, in the present work, we prove a class of modular equations in Ramanujan\u2019s alternative theory of elliptic functions using theory of signature 4. In particular, the composite degrees of modular equations {1, 2, 4}, {1, 3, 9}, {1, 5, 25} and {1, 7, 49}.\n                  <\/jats:p>","DOI":"10.1007\/s00500-023-08036-9","type":"journal-article","created":{"date-parts":[[2023,4,25]],"date-time":"2023-04-25T11:28:05Z","timestamp":1682422085000},"page":"16243-16249","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Note on modular equations in the theory of signature 4"],"prefix":"10.1007","volume":"27","author":[{"given":"B. R. Srivatsa","family":"Kumar","sequence":"first","affiliation":[]},{"given":"P. S.","family":"Guruprasad","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3043-9316","authenticated-orcid":false,"given":"N. V. Sayinath","family":"Udupa","sequence":"additional","affiliation":[]},{"family":"Raksha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,25]]},"reference":[{"key":"8036_CR1","first-page":"33","volume":"7","author":"C Adiga","year":"2003","unstructured":"Adiga C, Kim T, Naika MSM (2003) Modular equations in the theory of signature three and $$P$$-$$Q$$ identities. Adv Stud Contemp Math 7:33\u201340","journal-title":"Adv Stud Contemp Math"},{"key":"8036_CR2","doi-asserted-by":"publisher","first-page":"3975","DOI":"10.2298\/FIL1713975A","volume":"31","author":"C Adiga","year":"2017","unstructured":"Adiga C, Bulkhali NAS, Simsek Y, Srivastava HM (2017) A continued fraction of Ramanujan and some Ramanujan-Weber class invariants. Filomat 31:3975\u20133997","journal-title":"Filomat"},{"issue":"1","key":"8036_CR3","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1006\/jmaa.2001.7823","volume":"268","author":"ND Baruah","year":"2002","unstructured":"Baruah ND (2002) Modular equations for Ramanujan\u2019s cubic continued fraction. J Math Anal Appl 268(1):244\u2013255","journal-title":"J Math Anal Appl"},{"issue":"2","key":"8036_CR4","doi-asserted-by":"publisher","first-page":"270","DOI":"10.1016\/S0022-314X(02)00127-0","volume":"100","author":"ND Baruah","year":"2003","unstructured":"Baruah ND (2003) On some of Ramanujan\u2019s Schl$$\\ddot{a}$$fli-type \u201cmixed\u2019\u2019 modular equations. J Number Theory 100(2):270\u2013294","journal-title":"J Number Theory"},{"key":"8036_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0965-2","volume-title":"Ramanujan\u2019s Notebooks","author":"BC Berndt","year":"1991","unstructured":"Berndt BC (1991) Ramanujan\u2019s Notebooks. Springer-Verlag, Berlin"},{"key":"8036_CR6","first-page":"4163","volume":"347","author":"BC Berndt","year":"1995","unstructured":"Berndt BC, Bhargava S, Garvan FG (1995) Ramanujan\u2019s alternative theories of elliptic functions. Trans Am Math Soc 347:4163\u20134244","journal-title":"Trans Am Math Soc"},{"key":"8036_CR7","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1006\/jnth.2000.2615","volume":"88","author":"BC Berndt","year":"2001","unstructured":"Berndt BC, Chan HH, Liaw WC (2001) On Ramanujan\u2019s quartic theory of elliptic functions. J Number Theory 88:129\u2013156","journal-title":"J Number Theory"},{"issue":"1","key":"8036_CR8","first-page":"23","volume":"45","author":"S Bhargava","year":"2003","unstructured":"Bhargava S, Adiga C, Naika MSM (2003) A new class of modular equations in Ramanujan\u2019s alternative theory of elliptic functions of signature 4 and some new $$P$$-$$Q$$ eta-function identities. Indian J Math 45(1):23\u201339","journal-title":"Indian J Math"},{"key":"8036_CR9","first-page":"691","volume":"323","author":"JM Borwein","year":"1991","unstructured":"Borwein JM, Borwien PB (1991) A cubic counterpart of Jacobi\u2019s identity and the AGM. Trans Am Math Soc 323:691\u2013701","journal-title":"Trans Am Math Soc"},{"key":"8036_CR10","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1006\/jsco.1995.1063","volume":"20","author":"FG Garvan","year":"1995","unstructured":"Garvan FG (1995) Ramanujan\u2019s theories of elliptic functions to alternative bases-a symbolic excursion. J Symbolic Comput 20:517\u2013536","journal-title":"J Symbolic Comput"},{"key":"8036_CR11","first-page":"1","volume":"22","author":"MSM Naika","year":"2006","unstructured":"Naika MSM (2006) A note on cubic modular equation of degree two. Tamsui Oxford J Math Sci 22:1\u20138","journal-title":"Tamsui Oxford J Math Sci"},{"issue":"3","key":"8036_CR12","first-page":"585","volume":"51","author":"MSM Naika","year":"2009","unstructured":"Naika MSM, Denis RY, Bairy KS (2009) On some Ramanujan-Selberg continued fraction. Indian J Math 51(3):585\u2013596","journal-title":"Indian J Math"},{"key":"8036_CR13","unstructured":"Naika MSM, Chandan Kumar S (2010) Some new Schl\u00e4fli type modular equation of signature 4. Ramanujan Math. Soc Lect Notes Ser No 14:185\u2013199"},{"key":"8036_CR14","first-page":"350","volume":"45","author":"S Ramanujan","year":"1914","unstructured":"Ramanujan S (1914) Modular equations and approximation to $$\\pi $$, Quart. J Math (Oxford) 45:350\u2013372","journal-title":"J Math (Oxford)"},{"key":"8036_CR15","unstructured":"Ramanujan S (1957) Notebooks Vol. 1 and 2, Tata Institute of Fundamental Research, Bombay"},{"key":"8036_CR16","volume-title":"Collected Papers, Chelsea","author":"S Ramanujan","year":"1962","unstructured":"Ramanujan S (1962) Collected Papers, Chelsea. Springer, USA"},{"key":"8036_CR17","doi-asserted-by":"crossref","unstructured":"Saikia N, Chetry J (2019) Some new modular equations in Ramanujan\u2019s alternate theory of signature 3. Ramanujan J 50:163\u2013194","DOI":"10.1007\/s11139-018-0115-7"},{"issue":"2","key":"8036_CR18","first-page":"281","volume":"19","author":"SP Singh","year":"2011","unstructured":"Singh SP, Yadav Vijay (2011) On modular identities and evaluation of certain theta functions. Rev Bull Cal Math Soc 19(2):281\u2013290","journal-title":"Rev Bull Cal Math Soc"},{"issue":"9","key":"8036_CR19","doi-asserted-by":"publisher","first-page":"2847","DOI":"10.2298\/FIL2009847S","volume":"34","author":"BR Srivatsa Kumar","year":"2020","unstructured":"Srivatsa Kumar BR (2020) Shruthi New modular equations of signature three in the spirit of Ramanujan. Filomat 34(9):2847\u20132868","journal-title":"Filomat"},{"key":"8036_CR20","volume-title":"Development of elliptic functions according to Ramanujan, Technical Report 2","author":"K Venkatachaliengar","year":"1988","unstructured":"Venkatachaliengar K (1988) Development of elliptic functions according to Ramanujan, Technical Report 2. Madurai Kamaraj University, Madurai"},{"key":"8036_CR21","unstructured":"Yadav Vijay, Chandankumar S (2021) On mixed modular equations of degree 21. South East Asian J of Math Math Sci 17(1):153\u2013170"}],"updated-by":[{"DOI":"10.1007\/s00500-023-08411-6","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2023,5,11]],"date-time":"2023-05-11T00:00:00Z","timestamp":1683763200000}}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-023-08036-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-023-08036-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-023-08036-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T07:11:24Z","timestamp":1694761884000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-023-08036-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,25]]},"references-count":21,"journal-issue":{"issue":"21","published-print":{"date-parts":[[2023,11]]}},"alternative-id":["8036"],"URL":"https:\/\/doi.org\/10.1007\/s00500-023-08036-9","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,25]]},"assertion":[{"value":"11 March 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 April 2023","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 May 2023","order":3,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":4,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":5,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s00500-023-08411-6","URL":"https:\/\/doi.org\/10.1007\/s00500-023-08411-6","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This study does not involve any human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}