[1] Ahlgren S. The sixth, eighth, ninth and tenth powers of Ramanujan theta function. Proc Amer Math Soc, 2000, 128: 1333-1338
[2] Andrews G E, Berndt B C. Ramanujan's Lost Notebook. Part III. New York: Springer-Verlag, 2012
[3] Bellman R. A Brief Introduction to Theta Functions. New York: Holt, Rinehart and Winston, 1961
[4] Berndt B C. Ramanujan's Notebooks. Part III. New York: Springer-Verlag, 1991
[5] Berndt B C. Ramanujan's Notebooks. Part V. New York: Springer-Verlag, 1998
[6] Boon M, Glasser M L, Zak J, Zucker I J. Additive decompositions of v-functions of multiple arguments. J Phys A: Math Gen, 1982, 15: 3439-3440
[7] Borwein J M, Borwein P B. A cubic counterpart of Jacobi's identity and the AGM. Trans Amer Math Soc, 1991, 323: 691-701
[8] Cai Y, Chen S, Luo Q M. Some circular summation formulas for the theta functions. Bound Value Probl, DOI: 10.1186/1687-2770-2013-59
[9] Cai Y, Bi Y Q, Luo Q M. Some new circular summation formulas of theta functions. Integral Transform Spec Funct, 2014, 25: 497-507
[10] Zhou Y, Luo Q M. Circular summation formulas for theta function v4(z|τ). Adv Difference Equ, DOI: 10.1186/1687-1847-2014-243
[11] Liu X F, Luo Q M. A note for alternating Ramanujan's circular summation formula. Forum Mathematicum, DOI: 10 1515/FORUM.2013.0148
[12] Chan S H, Liu Z G. On a new circular summation of theta functions. J Number Theory, 2010, 130: 1190-1196
[13] Chan H H, Liu Z G, Ng S T. Circular summation of theta functions in Ramanujan's lost notebook. J Math Anal Appl, 2006, 316: 628-641
[14] Chandrasekharan K. Elliptic Functions. Berlin, Heidelberg: Springer-Verlag, 1985
[15] Chua K S. The root lattice An* and Ramanujan's circular summation of theta functions. Proc Amer Math Soc, 2002, 130: 1-8
[16] Chua K S. Circular summation of the 13th powers of Ramanujan's theta function. Ramanujan J, 2001, 5: 353-354
[17] Gasper G, Rahma M. Basic Hypergeometric Series. Cambridge: Cambridge Unversity Press, 2004
[18] Liu Z G. Some inverse relations and theta function identities. Int J Number Theory, 2012, 8: 1977-2002
[19] Montaldi E, Zucchelli G. Additive decomposition for the product of two v3 functions and modular equations. J Math Phys, 1989, 30: 2012-2015
[20] Murayama T. On an identity of theta functions obtained from weight enumerators of linear codes. Proc Japan Acad, Ser A, 1998, 74: 98-100
[21] Ono K. On the circular summation of the eleventh powers of Ramanujan's theta function. J Number Theory, 1999, 76: 62-65
[22] Ramanujan S. The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa, 1988
[23] Rangachari S S. On a result of Ramanujan on theta functions. J Number Theory, 1994, 48: 364-372
[24] Shen L C. On the additive formula of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5. Trans Amer Math Soc, 1994, 345: 323-345
[25] Son S H. Circular summation of theta functions in Ramanujan's lost notebook. Ramanujan J, 2004, 8: 235-272
[26] Whittaker E T, Watson G N. A Course of Modern Analysis. Cambridge: Cambridge University Press, 1927
[27] Xu P. An elementary proof of Ramanujan's circular summation formula and its generalizations. Ramanujan J, 2012, 27: 409-417
[28] Zeng X F. A generalized circular summation of theta function and its application. J Math Anal Appl, 2009, 356: 698-703
[29] Zhu J M. An alternate circular summation formula of theta functions and its applications. Appl Anal Discrete Math, 2012, 6: 114-125
[30] Zhu J M. A note on a generalized circular summation formula of theta functions. J Number Theory, 2012, 132: 1164-1169
|