본문 바로가기
KERD
논문KCI 등재

SOME MODULAR RELATIONS BETWEEN RAMANUJAN’S FUNCTION V(q) AND V(qn)

B. N. DHARMENDRA|M. C. MAHESH KUMAR
장전수학회 (Jangjeon Mathematical Society)|2020.07|Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집) (Proceedings of the Jangjeon Mathematical Society)|제23권 제3호|pp. 347-359|DOI 10.17777/pjms.2020.23.3.007

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

Continued fractionModular equationTheta function
주제분류수학

초록

In his ‘lost’ notebook, S. Ramanujan introduced the parameter k(q) :=R(q)R2(q2) and v(q) := R2(q½)R(q)/R(q2) related to the Rogers-Ramanujan continued fraction R(q). In this paper, we establish some new modular relations connecting k(q) with v(qn) and v(q) with v(qn) for n = 6, 8 and 10.

Abstract

In his ‘lost’ notebook, S. Ramanujan introduced the parameter k(q) :=R(q)R2(q2) and v(q) := R2(q½)R(q)/R(q2) related to the Rogers-Ramanujan continued fraction R(q). In this paper, we establish some new modular relations connecting k(q) with v(qn) and v(q) with v(qn) for n = 6, 8 and 10.

참고문헌 (23)

  1. C. Adiga, "On Ramnujan’s cubic continued fraction and explicit evaluations of theta-functions", Indian J. pure appl. math, 35(9), pp.1047~1062 (2004)
  2. C. Adiga, "Some new modular relations for the Roger-Ramanujan type functions of order eleven with applications to partitions", J. Number Theory, 158, pp.281~297 (2016)
  3. G. E. Andrews, "The continued fractions found in the unorganized portions of Ramanujan’s notebooks", Mem. Am. Math. Soc, 99(477) (1992)
  4. G. E. Andrews, "Ramanujan’s Lost Notebook, Part I" (2005), Springer
  5. A. Bayad, "Identities involving vales of Bernoulli, q-Bernoulli, and q-Euler polynomials", Russ. J. Math. Phys, 18(2), pp.133~143 (2011)
  6. N. D. Baruah, "Two parameters for Ramanujan’s theta–functions and their explicit values", Rocky Mountain J. Math, 37(6), pp.1747~1790 (2007)
  7. B. C. Berndt, "Ramanujan’s Notebooks, Part III" (1991), Springer-Verlag
  8. B. C. Berndt, "Some theorems on the Rogers-Ramanujan continued fraction in Ramanujan’s lost notebook", Trans. Am. Math. Soc, 352, pp.2157~2177 (2000)
  9. S. Cooper, "On Ramanujan’s function k(q) = r(q)r2(q2)", Ramanujan J, 20(3), pp.311~328 (2009)
  10. C. Gugg, "Two modular equations for squares of the Rogers-Ramanujan functions with applications", Ramanujan J, 18(2), pp.183~207 (2009)
  11. S. -Y. Kang, "Some theorems on the Rogers-Ramanujan continued fraction and associated theta function identities in Ramanujan’s lost notebook", Ramanujan J, 3(1), pp.91~11 (1999)
  12. M. S. Mahadeva Naika, "New identities for Ramanujan’s cubic continued fraction", Funct. Approx. Comment. Math., Funct. Approx. Comment. Math, 46(1), pp.29~44 (2012)
  13. M. S. Mahadeva Naika, "On some explicit evaluation of the ratios of Ramanujan’s theta–function", Bull. Allahabad Math. Soc, 29(1), pp.53~86 (2014)
  14. M. S. Mahadeva Naika, "New Modular Relations for Ramanujan Parameter(q)", Int. J. Pure Appl. Math, 74(4), pp.413~435 (2012)
  15. M. S. Mahadeva Naika, "On Some New Modular Relations for Ramanujan’s (q)-function and (q)-function", Palestine Journal of Math, 3(1), pp.44~60 (2014)
  16. S. Ramanujan, "Notebooks (2 volumes)" (1957), Tata Institute of Fundamental Research
  17. S. Ramanujan, "The lost notebook and other unpublished papers" (1988), Narosa
  18. L. J. Rogers, "Second memoir on the expansion of certain infinite products", Proc. London Math. Soc, 25, pp.318~343 (1894)
  19. L. J. Rogers, "On a type of modular relation", Proc. Lond. Math. Soc, 19, pp.387~397 (1921)
  20. H. M. Srivastava, "Some Relationships Between q-Product Identities, Combinatorial Partition Identities and Continued-Fraction Identities", Advanced Studies in Contemporary Mathematics, 25(3), pp.265~272 (2015), 장전수학회
  21. G. N. Watson, "Theorems stated by Ramanujan (VII): theorems on continued fractions", J. London Math. Soc, 4, pp.39~48 (1929)
  22. G. N. Watson, "Theorems stated by Ramanujan (IX): Two continued fractions", J. London Math. Soc, 4, pp.231~237 (1929)
  23. J. Yi, "The explicit formulas and evaluations of Ramanujan’s theta-function", J. Math. Anal. Appl, 321, pp.157~181 (2006)

관련 논문AI추천

New identities for ratios of Ramanujan’s theta functionK. Sushan Bairy, M. S. MAHADEVA NAIKA, S. Chandankumar장전수학회 · Advanced Studies in Contemporary Mathematics2017 · 제27권 제1호Certain identities for a continued fraction of RamanujanK. S. Bairy, M. S. M. Naika, S. Chandankumar장전수학회 · Advanced Studies in Contemporary Mathematics2014 · 제24권 제1호Special Case on Rogers-Ramanujan Type Continued Fraction IdentityB. N. DHARMENDRA장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2020 · 제23권 제3호General Formulas for Explicit Evaluations of Ramanujan’s Cubic Continued FractionK. Sushan Bairy, M. C. Maheshkumar, M. S. Mahadeva Naika경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2009 · 제49권 제3호Identities for the Rogers-Ramanujan continued fractionNayandeep Deka Baruah, Pranjal Talukdar대한수학회 · 대한수학회지2025 · 제62권 제1호Some more relations on Ramanujan's cubic continued fraction and a continued fraction of order 12B. R. S. Kumar, H. C. Vidya장전수학회 · Advanced Studies in Contemporary Mathematics2014 · 제24권 제2호

이 간행물의 다른 논문 (13건)

LOCALLY BOUNDED NOT A PRIORI CONTINUOUS HOMOMORPHISMS OF NONCOMPACT SEMISIMPLE LIE GROUPS INTO CONNECTED AMENABLE LIE GROUPS ARE IDENTITY HOMOMORPHISMSA. I. SHTERNpp. 293-296THE GROWTH RATE OF RANDOM BALANCING SEQUENCEA. Patra, G. K. Panda, T KOMATSUpp. 297-304ON SOME P-Q MIXED MODULAR EQUATIONS OF DEGREE 5M. S. MAHADEVA NAIKA, S. Chandankumar, M. Harishpp. 305-319New Modular Relations for the Ratio’s of Ramanujan Function χ(−q) of degree 9P. S. Guruprasad, B. N. DHARMENDRApp. 321-329Special Case on Rogers-Ramanujan Type Continued Fraction IdentityB. N. DHARMENDRApp. 331-340
원문 보기