On combinatorial extensions of some Ramanujan’s mock theta functions

  • M. Goyal IK Gujral Punjab Techn. Univ., Jalandhar, India
Keywords: Ramanujan’s mock theta functions, mock theta functions, $(n t)$--color partitions, weighted lattice paths, associated lattice paths, anti-hook differences


Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path functions and antihook differences. These results provide new combinatorial interpretations of five mock theta functions of Ramanujan.
Using a bijection between the associated lattice path functions and the $(n+t)$-color partitions and then between the associated lattice path functions and the weighted lattice path functions, we extend the works by Agarwal and Agarwal and Rana to five new 3-way combinatorial identities. These results are further extended to 4-way combinatorial identities by using bijection between the $(n+t)$-color partitions and the partitions with certain antihook differences. These interesting results present elegant combinatorial links between Ramanujan's mock theta functions, $(n+t)$-color partitions, weighted lattice paths, associated lattice paths, and antihook differences.



Agarwal, A. K. Rogers-Ramanujan identities for $n$-color partitions. J. Number Theory 28 (1988), no. 3, 299--305. doi: 10.1016/0022-314X(88)90045-5

Agarwal, A. K. Antihook differences and some partition identities. Proc. Amer. Math. Soc. 110 (1990), no. 4, 1137--1142. doi: 10.2307/2047768

A. K. Agarwal, Lattice paths and Rogers – Ramanujan type identities, Proc. 14th annual Conf. of the Ramanujan Math. Soc., Banagalore, 31 – 39 (1999).

Agarwal, A. K. $n$-color partition theoretic interpretations of some mock theta functions. Electron. J. Combin. 11 (2004), no. 1, Note 14, 6 pp. doi: 10.37236/1855

A. K. Agarwal, Lattice paths and mock theta functions, Proc. 6th Int. Conf. SSFA, Jaunpur, India, 95 – 102 (2005).

Agarwal, A. K.; Andrews, G. E. Hook differences and lattice paths. J. Statist. Plann. Inference 14 (1986), no. 1, 5--14. doi: 10.1016/0378-3758(86)90004-2

Agarwal, A. K.; Andrews, George E. Rogers-Ramanujan identities for partitions with "$n$ copies of $n$''. J. Combin. Theory Ser. A 45 (1987), no. 1, 40--49. doi: 10.1016/0097-3165(87)90045-8

Agarwal, A. K.; Bressoud, David M. Lattice paths and multiple basic hypergeometric series. Pacific J. Math. 136 (1989), no. 2, 209--228. MR0978611

Agarwal, Ashok Kumar; Goyal, Megha. Lattice paths and Rogers identities. Open J. Discrete Math. 1 (2011), no. 2, 89--95. doi: 10.4236/ojdm.2011.12011

Agarwal, A. K.; Goyal, M. New partition theoretic interpretations of Rogers-Ramanujan identities. Int. J. Comb. 2012, Art. ID 409505, 6 pp. doi: 10.1155/2012/409505

Agarwal, A. K.; Goyal, Megha. On 3-way combinatorial identities. Proc. Indian Acad. Sci. Math. Sci. 128 (2018), no. 1, Art. 2, 21 pp. doi: 10.1007/s12044-018-0378-3

Agarwal, A. K.; Rana, M. Two new combinatorial interpretations of a fifth order mock theta function. J. Indian Math. Soc. (N.S.) 2007, Special volume on the occasion of the centenary year of IMS (1907-2007), 11--24 (2008). MR2518230

Weisner, Louis. Recent Publications: Reviews: Collected Papers of Srinivasa Ramanujan. Amer. Math. Monthly 35 (1928), no. 6, 320--321. doi: 10.2307/2298686

Anand, S.; Agarwal, A. K. A new class of lattice paths and partitions with $n$ copies of $n$. Proc. Indian Acad. Sci. Math. Sci. 122 (2012), no. 1, 23--39. doi: 10.1007/s12044-012-0057-8

Andrews, George E. Generalized Frobenius partitions. Mem. Amer. Math. Soc. 49 (1984), no. 301, {rm iv}+44 pp. doi: 10.1090/memo/0301

Andrews, George E.; Hickerson, Dean. Ramanujan's "lost'' notebook. VII. The sixth order mock theta functions. Adv. Math. 89 (1991), no. 1, 60--105. doi: 10.1016/0001-8708(91)90083-J

S. Bhargava, S.; Vasuki, K. R.; Rajanna, K. R. On some Ramanujan identities for the ratios of eta-functions. Reprint of Ukraïn. Mat. Zh. 66 (2014), no. 8, 1011–1028. Ukrainian Math. J. 66 (2015), no. 8, 1131--1151. doi: 10.1007/s11253-015-0999-y

Choi, Youn-Seo. Tenth order mock theta functions in Ramanujan's lost notebook. Invent. Math. 136 (1999), no. 3, 497--569. doi: 10.1007/s002220050318

Chu, W.; Wang, C. Iteration process for multiple Rogers-Ramanujan identities. Ukrainian Math. J. 64 (2012), no. 1, 110--139. doi: 10.1007/s11253-012-0633-1

Gordon, Basil; McIntosh, Richard J. Some eighth order mock theta functions. J. London Math. Soc. (2) 62 (2000), no. 2, 321--335. doi: 10.1112/S0024610700008735

Gordon, Basil; McIntosh, Richard J. Modular transformations of Ramanujan's fifth and seventh order mock theta functions. Rankin memorial issues. Ramanujan J. 7 (2003), no. 1-3, 193--222. doi: 10.1023/A:1026299229509

Goyal, M.; Agarwal, A. K. Further Rogers-Ramanujan identities for $n$-color partitions. Util. Math. 95 (2014), 141--148. MR3243926

Goyal, M.; Agarwal, A. K. On a new class of combinatorial identities. Ars Combin. 127 (2016), 65--77. doi: 10.4236/ojdm.2014.44012

Goyal, Megha. New combinatorial interpretations of some Rogers-Ramanujan type identities. Contrib. Discrete Math. 11 (2017), no. 2, 43--57. doi: 10.11575/cdm.v11i2.62576

Goyal, Megha. On combinatorial extensions of Rogers-Ramanujan type identities. Contrib. Discrete Math. 12 (2017), no. 2, 33--51. doi: 10.11575/cdm.v12i2.62496

McIntosh, Richard J. Second order mock theta functions. Canad. Math. Bull. 50 (2007), no. 2, 284--290. doi: 10.4153/CMB-2007-028-9

McIntosh, Richard J. The $H$ and $K$ family of mock theta functions. Canad. J. Math. 64 (2012), no. 4, 935--960. doi: 10.4153/CJM-2011-066-0

How to Cite
Goyal, M. “On Combinatorial Extensions of Some Ramanujan’s Mock Theta Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 1, Jan. 2020, pp. 46-60, http://umj.imath.kiev.ua/index.php/umj/article/view/2327.
Research articles