SIGMA 22 (2026), 038, 12 pages      arXiv:2510.17209      https://doi.org/10.3842/SIGMA.2026.038

On Bilateral Multiple Sums and Rogers-Ramanujan Type Identities

Dandan Chen ^ab and Tianjian Xu ^a
a) Department of Mathematics, Shanghai University, P.R. China
^b) Newtouch Center for Mathematics, Shanghai University, P.R. China

Received October 21, 2025, in final form March 29, 2026; Published online April 18, 2026

Abstract
We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic hypergeometric series in conjunction with the integral method.

Key words: Rogers-Ramanujan type identities; bilateral summations; multiple sums; $q$nobreakdash-series; integral method.

pdf (401 kb)   tex (17 kb)  

References

1. Andrews G.E., On the general Rogers-Ramanujan theorem, Mem. Amer. Math. Soc., Vol. 152, American Mathematical Society, Providence, RI, 1974.
2. Andrews G.E., Multiple $q$-series identities, Houston J. Math. 7 (1981), 11-22.
3. Berndt B.C., Ramanujan's notebooks. Part III, Springer, New York, 1991.
4. Bressoud D.M., Analytic and combinatorial generalizations of the Rogers-Ramanujan identities, Mem. Amer. Math. Soc. 24 (1980), 54 pages.
5. Cao Z., Wang L., Multi-sum Rogers-Ramanujan type identities, J. Math. Anal. Appl. 522 (2023), 126960, 24 pages, arXiv:2205.12786.
6. Cho B., Koo J.K., Park Y.K., Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction, J. Number Theory 129 (2009), 922-947.
7. Frye J., Garvan F., Automatic proof of theta-function identities, in Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Texts Monogr. Symbol. Comput., Springer, Cham, 2019, 195-258, arXiv:1807.08051.
8. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia Math. Appl., Vol. 96, Cambridge University Press, Cambridge, 2004.
9. Hardy G.H., The Indian Mathematician Ramanujan, Amer. Math. Monthly 44 (1937), 137-155.
10. Nahm W., Conformal field theory, dilogarithms, and three-dimensional manifolds, Adv. Appl. Clifford Algebras 4 (1994), suppl. 1, 179-191.
11. Nahm W., Conformal field theory and the dilogarithm, in XIth International Congress of Mathematical Physics (Paris, 1994), International Press, Cambridge, MA, 1995, 662-667.
12. Nahm W., Conformal field theory and torsion elements of the Bloch group, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 67-132, arXiv:hep-th/0404120.
13. Rankin R.A., Modular forms and functions, Cambridge University Press, Cambridge, 1977.
14. Robins S., Generalized Dedekind $\eta$-products, in The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemp. Math., Vol. 166, American Mathematical Society, Providence, RI, 1994, 119-128.
15. Rogers L.J., Second memoir on the expansion of certain infinite products, Proc. Lond. Math. Soc. 25 (1893/94), 318-343.
16. Schlosser M.J., Bilateral identities of the Rogers-Ramanujan type, Trans. Amer. Math. Soc. Ser. B 10 (2023), 1119-1140, arXiv:1806.01153.
17. Slater L.J., Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. 54 (1952), 147-167.
18. Wang L., New proofs of some double sum Rogers-Ramanujan type identities, Ramanujan J. 62 (2023), 251-272, arXiv:2203.15572.
19. Wang L., Explicit forms and proofs of Zagier's rank three examples for Nahm's problem, Adv. Math. 450 (2024), 109743, 46 pages, arXiv:2211.04375.
20. Wang L., Identities on Zagier's rank two examples for Nahm's problem, Res. Math. Sci. 11 (2024), 49, 34 pages, arXiv:2210.10748.
21. Warnaar S.O., Zudilin W., Dedekind's $\eta$-function and Rogers-Ramanujan identities, Bull. Lond. Math. Soc. 44 (2012), 1-11, arXiv:1001.1571.
22. Zagier D., The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 3-65.