Geodesic
On a Transformation of Triple $q$-Series and Rogers–Hecke Type Series
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Using the method of the $q$-exponential differential operator, we give an extension of the Sears $_4\phi_3$ transformation formula. Based on this extended formula and a $q$-series expansion formula for an analytic function around the origin, we present a transformation formula for triple $q$-series, which includes several interesting special cases, especially a double $q$-series summation formula. Some applications of this transformation formula to Rogers–Hecke type series are discussed. More than 100 Rogers–Hecke type identities including Andrews' identities for the sums of three squares and the sums of three triangular numbers are obtained.
Keywords: $q$-partial differential equation, double $q$-series summation, triple $q$-hypergeometric series, $q$-exponential differential operator, Rogers–Hecke type series 
@article{SIGMA_2024_20_a85,
     author = {Zhi-Guo Liu},
     title = {On a {Transformation} of {Triple} $q${-Series} and {Rogers{\textendash}Hecke} {Type} {Series}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
     volume = {20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a85/}
}
TY  - JOUR
AU  - Zhi-Guo Liu
TI  - On a Transformation of Triple $q$-Series and Rogers–Hecke Type Series
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2024
VL  - 20
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a85/
LA  - en
ID  - SIGMA_2024_20_a85
ER  - 
%0 Journal Article
%A Zhi-Guo Liu
%T On a Transformation of Triple $q$-Series and Rogers–Hecke Type Series
%J Symmetry, integrability and geometry: methods and applications
%D 2024
%V 20
%U http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a85/
%G en
%F SIGMA_2024_20_a85
Zhi-Guo Liu. On a Transformation of Triple $q$-Series and Rogers–Hecke Type Series. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a85/

[1] Andrews G.E., “Enumerative proofs of certain $q$-identities”, Glasgow Math. J., 8 (1967), 33–40 | DOI

[2] Andrews G.E., “Applications of basic hypergeometric functions”, SIAM Rev., 16 (1974), 441–484 | DOI

[3] Andrews G.E., “The fifth and seventh order mock theta functions”, Trans. Amer. Math. Soc., 293 (1986), 113–134 | DOI

[4] Andrews G.E., “$q$-orthogonal polynomials, Rogers–Ramanujan identities, and mock theta functions”, Proc. Steklov Inst. Math., 276 (2012), 21–32 | DOI

[5] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia Math. Appl., 71, Cambridge University Press, Cambridge, 1999 | DOI

[6] Andrews G.E., Berndt B.C., Ramanujan's lost notebook, v. I, Springer, New York, 2005 | DOI

[7] Andrews G.E., Dyson F.J., Hickerson D., “Partitions and indefinite quadratic forms”, Invent. Math., 91 (1988), 391–407 | DOI

[8] Aslan H., Ismail M.E.H., “A $q$-translation approach to Liu's calculus”, Ann. Comb., 23 (2019), 465–488 | DOI

[9] Chen D., Wang L., “Representations of mock theta functions”, Adv. Math., 365 (2020), 107037, 72 pp., arXiv: 1811.07686 | DOI

[10] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia Math. Appl., 96, 2nd ed., Cambridge University Press, Cambridge, 2004 | DOI

[11] Jackson F., “On $q$-functions and a certain difference operator”, Trans. Roy. Soc. Edin., 46 (1909), 253–281 | DOI

[12] Krammer D., “Sums of three squares and $q$-series”, J. Number Theory, 44 (1993), 244–254 | DOI

[13] Liu Z.-G., “An expansion formula for $q$-series and applications”, Ramanujan J., 6 (2002), 429–447 | DOI

[14] Liu Z.-G., “Some operator identities and $q$-series transformation formulas”, Discrete Math., 265 (2003), 119–139 | DOI

[15] Liu Z.-G., “An extension of the quintuple product identity and its applications”, Pacific J. Math., 246 (2010), 345–390 | DOI

[16] Liu Z.-G., “Two $q$-difference equations and $q$-operator identities”, J. Difference Equ. Appl., 16 (2010), 1293–1307 | DOI

[17] Liu Z.-G., “An extension of the non-terminating ${}_6\phi_5$ summation and the Askey–Wilson polynomials”, J. Difference Equ. Appl., 17 (2011), 1401–1411 | DOI

[18] Liu Z.-G., “A $q$-series expansion formula and the Askey–Wilson polynomials”, Ramanujan J., 30 (2013), 193–210 | DOI

[19] Liu Z.-G., “On the $q$-derivative and $q$-series expansions”, Int. J. Number Theory, 9 (2013), 2069–2089, arXiv: 1805.04618 | DOI

[20] Liu Z.-G., “On the $q$-partial differential equations and $q$-series”, The Legacy of Srinivasa Ramanujan, Ramanujan Math. Soc. Lect. Notes Ser., 20, Ramanujan Mathematical Society, Mysore, 2013, 213–250, arXiv: 1805.02132

[21] Liu Z.-G., “A $q$-extension of a partial differential equation and the Hahn polynomials”, Ramanujan J., 38 (2015), 481–501, arXiv: 1805.07292 | DOI

[22] Liu Z.-G., “On a system of $q$-partial differential equations with applications to $q$-series”, Analytic Number Theory, Modular Forms and $q$-hypergeometric Series, Springer Proc. Math. Stat., 221, Springer, Cham, 2017, 445–461, arXiv: 1709.06784 | DOI

[23] Liu Z.-G., “Askey–Wilson polynomials and a double $q$-series transformation formula with twelve parameters”, Proc. Amer. Math. Soc., 147 (2019), 2349–2363, arXiv: 1810.02918 | DOI

[24] Liu Z.-G., A universal identity for theta functions of degree eight and applications, Hardy-Ramanujan J., 43 (2020), 129–172, arXiv: 2104.14705 | DOI

[25] Liu Z.-G., “On the Askey–Wilson polynomials and a $q$-beta integral”, Proc. Amer. Math. Soc., 149 (2021), 4639–4648 | DOI

[26] Liu Z.-G., “A multiple $q$-exponential differential operational identity”, Acta Math. Sci. Ser. B (Engl. Ed.), 43 (2023), 2449–2470 | DOI

[27] Liu Z.-G., “A multiple $q$-translation formula and its implications”, Acta Math. Sin. (Engl. Ser.), 39 (2023), 2338–2363 | DOI

[28] Schendel L., “Zur Theorie der Functionen”, J. Reine Angew. Math., 84 (1878), 80–84 | DOI

[29] Shanks D., “A short proof of an identity of Euler”, Proc. Amer. Math. Soc., 2 (1951), 747–749 | DOI

[30] Shanks D., “Two theorems of Gauss”, Pacific J. Math., 8 (1958), 609–612 | DOI

[31] Wang C., Chern S., “Some $q$-transformation formulas and Hecke type identities”, Int. J. Number Theory, 15 (2019), 1349–1367 | DOI

[32] Wang L., Yee A.J., “Some Hecke–Rogers type identities”, Adv. Math., 349 (2019), 733–748 | DOI