Notes on $q$-Partial Differential Equations for $q$-Laguerre Polynomials and Little $q$-Jacobi Polynomials

Year 2024, Volume: 7 Issue: 2, 59 - 76, 30.06.2024

Qi Bao , Dunkun Yang

https://doi.org/10.33401/fujma.1365120

Cited By: 1

Abstract

This article defines two common $q$-orthogonal polynomials: homogeneous $q$-Laguerre polynomials and homogeneous little $q$-Jacobi polynomials. They can be viewed separately as solutions to two $q$-partial differential equations. Furthermore, an analytic function satisfies a certain system of $q$-partial differential equations if and only if it can be expanded in terms of homogeneous $q$-Laguerre polynomials or homogeneous little $q$-Jacobi polynomials. As applications, several generalized Ramanujan $q$-beta integrals and Andrews-Askey integrals are obtained.

Keywords

$q$-Laguerre polynomial, little $q$-Jacobi polynomial, $q$-partial differential equations, generating function, Ramanujan $q$-beta integrals, $q$-integrals

References

• [1] W.A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30(1-2) (1965), 47–61. $ \href{ https://doi.org/10.1002/mana.19650300105}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84966219999&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+Orthogonal+q-Polynomials%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=1}{[\mbox{Scopus}]} $
• [2] A. deMedicis and X.G. Viennot, Moments of Laguerre q-polynomials and the Foata-Zeilberger bijection, (French) Adv. in Appl. Math. 15(3) (1994), 262–304.
• [3] H.E. Heine, Handbuch der Kugelfunctionen, Theorie und Anwendungen, 2nd ed., 1, G. Reimer, Berlin, 1878.
• [4] N.N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York, (1972).
• [5] M.E.H. Ismail, A brief review of q-series, Lectures on orthogonal polynomials and special functions, 76–130, London Math. Soc. Lecture Note Ser., 464, Cambridge Univ. Press, Cambridge, (2021). $ \href{https://doi.org/10.1017/9781108908993.003}{[\mbox{CrossRef}]} $
• [6] G. Gasper and M. Rahman, Basic Hypergeometric Series, With a Foreword by Richard Askey, Second edition. Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004. $ \href{https://doi.org/10.1017/CBO9780511526251}{[\mbox{CrossRef}]} $
• [7] Z.G. Liu, A q-operational equation and the Rogers-Szeg˝o polynomials, Sci. China Math., 66(6) (2023), 1199–1216. $ \href{https://doi.org/10.1007/s11425-021-1999-2}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85146154571&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+%24q%24-operational+equation+and+the+Rogers-Szeg%C3%B6+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000912065500001}{[\mbox{Web of Science}]} $
• [8] Z.G. Liu, On the Askey-Wilson polynomials and a q-beta integral, Proc. Amer. Math. Soc., 149(11) (2021), 4639–4648. $ \href{http://dx.doi.org/10.1090/proc/15584}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85114845014&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+Askey-Wilson+polynomials+and+a+%24q%24-beta+integral%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000695492700010}{[\mbox{Web of Science}]} $
• [9] H.S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and ”q”) multisum/integral identities, Invent. Math., 108(3)(1992), 575–633. $ \href{https://doi.org/10.1007/BF02100618}{[\mbox{CrossRef}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1992HW22800004}{[\mbox{Web of Science}]} $
• [10] W.C. Chu, Inversion techniques and combinatorial identities, Boll. Un. Mat. Ital. B 7B(4) (1993), 737–760. $ \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1993MN83000001}{[\mbox{Web of Science}]} $
• [11] J. Gu, D.K. Yang and Q. Bao, Two q-Operational Equations and Hahn Polynomials, Complex Anal. Oper. Theory 18(3) (2024), Paper No. 53. $ \href{https://doi.org/10.1007/s11785-024-01496-3}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85187110474&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Two+%24q%24-Operational+Equations+and+Hahn+Polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001179086800002}{[\mbox{Web of Science}]} $
• [12] H.C. Agrawal and A.K. Agrawal, Basic hypergeometric series and the operator (qaD), J. Indian Acad. Math. 15(1)(1993), 81–88.
• [13] R. Askey and D.T. Haimo, Series inversion of some convolution transforms, J. Math. Anal. Appl. 59(1) (1977), 119–129. $ \href{https://doi.org/10.1016/0022-247X(77)90096-8}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-49449124969&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Series+inversion+of+some+convolution+transforms%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=0}{[\mbox{Scopus}]} $
• [14] W.Y.C. Chen and Z.G. Liu, Parameter augmentation for basic hypergeometric series, II, J. Combin. Theory Ser. A, 80(2) (1997), 175–195. $ \href{https://doi.org/10.1006/jcta.1997.2801}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0038877325&origin=resultslist&sort=plf-f&src=s&sid=091c2ace3e7c91bf2f9e1975aebaa5be&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Parameter+augmentation+for+basic+hypergeometric+series%22%29&sl=66&sessionSearchId=091c2ace3e7c91bf2f9e1975aebaa5be&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1997YH17700001}{[\mbox{Web of Science}]} $
• [15] W.Y.C. Chen and Z.G. Liu, Parameter Augmentation for Basic Hypergeometric Series, I. Mathematical Essays in Honor of Gian-Carlo Rota, 111–129, Progr. Math., 161, Birkhauser Boston, Boston, MA, (1998). $ \href{https://doi.org/10.1007/978-1-4612-4108-9_5}{[\mbox{CrossRef}]} $
• [16] G.E. Andrews, On a transformation of bilateral series with applications, Proc. Amer. Math. Soc. 25(3) (1970), 554–558. $ \href{https://doi.org/10.2307/2036642}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84968478823&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+transformation+of+bilateral+series+with+applications%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=0}{[\mbox{Scopus}]} $
• [17] M.J. Wang, q-integral representation of the Al-Salam-Carlitz polynomials, Appl. Math. Lett., 22(6)(2009), 943–945. $ \href{https://doi.org/10.1016/j.aml.2009.01.002}{[\mbox{CrossRef}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000266213800025}{[\mbox{Web of Science}]} $
• [18] J. Cao, Some integrals involving q-Laguerre polynomials and applications, Abstr. Appl. Anal. 2013, Art. ID 302642, 13 pp. $ \href{https://doi.org/10.1155/2013/302642}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84881505035&origin=resultslist&sort=plf-f&src=s&sid=091c2ace3e7c91bf2f9e1975aebaa5be&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+integrals+involving+%24q%24-Laguerre+polynomials+and+applications%22%29&sl=66&sessionSearchId=091c2ace3e7c91bf2f9e1975aebaa5be&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000322485200001}{[\mbox{Web of Science}]} $
• [19] H.L. Saad and M.A. Abdlhusein, New application of the Cauchy operator on the homogeneous Rogers-Szegö polynomials, Ramanujan J. 56(1)(2021), 347–367. $\href{https://doi.org/10.1007/s11139-021-00432-9}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85105951132&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22New+application+of+the+Cauchy+operator+on+the+homogeneous%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000650514000001}{[\mbox{Web of Science}]} $
• [20] H. Aslan and M.E.H. Ismail, A q-translation approach to Liu’s calculus, Ann. Comb., 23(3-4) (2019), 465–488. $ \href{https://doi.org/10.1007/s00026-019-00450-x}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85074590241&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22+A+%24q%24-translation+approach+to+Liu%27s+calculus%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=1}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000494203500001}{[\mbox{Web of Science}]} $
• [21] M.E.H. Ismail, R.M. Zhang and K. Zhou, q-fractional Askey-Wilson integrals and related semigroups of operators, Phys. D: Nonlinear Phenom., 442 (2022), Paper No. 133534, 15 pp. $ \href{https://doi.org/10.1016/j.physd.2022.133534}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85139855414&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22%24q%24-fractional+Askey-Wilson+integrals+and+related+semigroups+of+operators%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000880401700017}{[\mbox{Web of Science}]} $
• [22] W.C. Chu and J.M. Campbell, Expansions over Legendre polynomials and infinite double series identities, Ramanujan J., 60(2) (2023), 317–353. $\href{https://doi.org/10.1007/s11139-022-00663-4}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85145208425&origin=resultslist&sort=plf-f&src=s&sid=091c2ace3e7c91bf2f9e1975aebaa5be&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Expansions+over+Legendre+polynomials+and+infinite+double+series+identities%22%29&sl=66&sessionSearchId=091c2ace3e7c91bf2f9e1975aebaa5be&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000905912000004}{[\mbox{Web of Science}]} $
• [23] G. Bhatnagar and S. Rai, Expansion formulas for multiple basic hypergeometric series over root systems, Adv. in Appl. Math., 137 (2022), Paper No. 102329. $ \href{https://doi.org/10.1016/j.aam.2022.102329}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85126397158&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Expansion+formulas+for+multiple+basic+hypergeometric+series+over+root+systems%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000796004000001}{[\mbox{Web of Science}]} $
• [24] C.A. Wei and D.X. Gong, Several transformation formulas for basic hypergeometric series. J. Diff. Equ. Appl., 27(2) (2021), 157–171. $\href{https://doi.org/10.1080/10236198.2021.1876683}{[\mbox{CrossRef}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000611665700001}{[\mbox{Web of Science}]} $
• [25] J.P. Fang and V.J.W. Guo, Some q-supercongruences related to Swisher’s (H.3) conjecture, Int. J. Number Theory 18(7) (2022), 1417–1427. $ \href{https://doi.org/10.1142/S1793042122500725}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85122022625&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+%24q%24-supercongruences+related+to+Swisher%27s+%28H.3%29+conjecture%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000810484600003}{[\mbox{Web of Science}]} $
• [26] Z.G. Liu, Two q-difference equations and q-operator identities, J. Differ. Equ. Appl., 16(11) (2010), 1293–1307. $ \href{https://doi.org/10.1080/10236190902810385}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-78149366394&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Two+%24q%24-difference+equations+and+%24q%24-operator+identities%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000283879100004}{[\mbox{Web of Science}]} $
• [27] Z.G. Liu, An extension of the non-terminating 6f5 summation and the Askey-Wilson polynomials, J. Differ. Equ. Appl. 17(10) (2011), 1401– 1411. $ \href{https://doi.org/10.1080/10236190903530735}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-80053013753&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22summation+and+the+Askey-Wilson+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000295120800001}{[\mbox{Web of Science}]} $
• [28] Z.G. Liu, On the q-partial differential equations and q-series, in: The Legacy of Srinivasa Ramanujan, in: Ramanujan Math. Soc. Lect. Notes Ser., 20, Ramanujan Math. Soc., Mysore, (2013), 213–250. $\href{https://doi.org/10.48550/arXiv.1805.02132}{[\mbox{CrossRef}]} $
• [29] Z.G. Liu, A q-extension of a partial differential equation and the Hahn polynomials, Ramanujan J., 38(3) (2015), 481–501. $ \href{https://doi.org/10.1007/s11139-014-9632-1}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84947128473&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+%24q%24-extension+of+a+partial+differential+equation+and+the+Hahn+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000364969700003}{[\mbox{Web of Science}]} $
• [30] Z.G. Liu, On a system of partial differential equations and the bivariate Hermite polynomials, J. Math. Anal. Appl., 45(1) (2017), 1–17. $\href{https://doi.org/10.1016/j.jmaa.2017.04.066}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85018722466&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+system+of+partial+differential+equations+and+the+bivariate+Hermite+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000402450900001}{[\mbox{Web of Science}]} $
• [31] Z.G. Liu, On the complex Hermite polynomials, Filomat, 34(2) (2020), 409–420. $ \href{https://doi.org/10.2298/FIL2002409L}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85096915507&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+complex+Hermite+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000595329700014}{[\mbox{Web of Science}]} $
• [32] Z.G. Liu, A multiple q-translation formula and its implications, Acta Math. Sin., 39(12) (2023), 2338–2363. $ \href{https://doi.org/10.1007/s10114-023-2237-0}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85170224261&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+multiple+%24q%24-translation+formula+and+its+implications%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001071738200016}{[\mbox{Web of Science}]} $
• [33] Z.G. Liu, A multiple q-exponential differential operational identity, Acta Math. Sci. Ser. B, 43(6)(2023), 2449–2470. $ \href{https://doi.org/10.1007/s10473-023-0608-3}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85175813592&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+multiple+%24q%24-exponential+differential+operational+identity%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=1}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001103749200012}{[\mbox{Web of Science}]} $
• [34] D.W. Niu and L. Li, q-Laguerre polynomials and related q-partial differential equations, J. Diff. Equ. Appl. 24(3) (2018), 375–390. $\href{https://doi.org/10.1080/10236198.2017.1409221}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85037973429&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22%24q%24-Laguerre+polynomials+and+related+%24q%24-partial+differential+equations%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000423855600004}{[\mbox{Web of Science}]} $
• [35] J. Cao, A note on generalized q-difference equations for q-beta and Andrews-Askey integral, J. Math. Anal. Appl. 412(2) (2014), 841–851. $\href{https://doi.org/10.1016/j.jmaa.2013.11.027}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84891502262&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+note+on+generalized+%24q%24-difference+equations+for+%24q%24-beta+and+Andrews-Askey+integral%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000330419500020}{[\mbox{Web of Science}]} $
• [36] J. Cao, Homogeneous q-difference equations and generating functions for q-hypergeometric polynomials, Ramanujan J. 40(1) (2016), 177– 192. $ \href{http://dx.doi.org/10.1007%2Fs11139-015-9676-x}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84923338102&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Homogeneous+%24q%24-difference+equations+and+generating+functions+for+%24q%24-hypergeometric+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} $
• [37] J. Cao, Homogeneous q-partial difference equations and some applications, Adv. Appl. Math. 84 (2017), 47–72. $ \href{https://doi.org/10.1016/j.aam.2016.11.001}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84998953943&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Homogeneous+%24q%24-partial+difference+equations+and+some+applications%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000393264600004}{[\mbox{Web of Science}]} $
• [38] J. Cao and D.W. Niu, A note on q-difference equations for Cigler’s polynomials, J. Diff. Equ. Appl. 22(12) (2016), 1880–1892. $ \href{https://doi.org/10.1080/10236198.2016.1250750}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84994175603&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+note+on+%24q%24-difference+equations+for+Cigler%27s+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=1}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000393123900007}{[\mbox{Web of Science}]} $
• [39] S. Arjika, q-difference equations for homogeneous q-difference operators and their applications, J. Difference Equ. Appl. 26(7) (2020), 987–999. $ \href{https://doi.org/10.1080/10236198.2020.1804888}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85089467901&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22%24q%24-difference+equations+for+homogeneous+%24q%24-difference+operators+and+their+applications%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000559864400001}{[\mbox{Web of Science}]} $
• [40] M.A. Abdlhusein, Two operator representations for the trivariate q-polynomials and Hahn polynomials, Ramanujan J., 40(3) (2016), 491– 509. $ \href{https://doi.org/10.1007/s11139-015-9731-7}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84946882220&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Two+operator+representations+for+the+trivariate%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000379866300004}{[\mbox{Web of Science}]} $
• [41] J. Cao, T.X. Cai and L.P. Cai, A note on q-partial differential equations for generalized q-2D Hermite polynomials, Progress on difference equations and discrete dynamical systems, 201–211, Springer Proc. Math. Stat., 341, Springer, Cham, (2020). $ \href{https://doi.org/10.1007/978-3-030-60107-2_8}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85101536544&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+note+on+%24q%24-partial+differential+equations+for+generalized+%24q%24-2D+Hermite+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} $
• [42] J. Cao, H.L. Zhou and S. Arjika, Generalized homogeneous q-difference equations for q-polynomials and their applications to generating functions and fractional q-integrals, Adv. Differ. Equ., 2021(1) (2021), 329. $ \href{https://doi.org/10.1186/s13662-021-03484-9}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85109789708&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Generalized+homogeneous+%24q%24-difference+equations+for+%24q%24-polynomials+and+their+applications+to+generating+functions+and+fractional+%24q%24-integrals%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000672668900002}{[\mbox{Web of Science}]} $
• [43] Z.Y. Jia, Homogeneous q-difference equations and generating functions for the generalized 2D-Hermite polynomials, Taiwanese J. Math. 25(1) (2021), 45–63. $\href{https://doi.org/10.11650/tjm/200804}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85100784193&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Homogeneous+%24q%24-difference+equations+and+generating+functions+for+the+generalized+2D-Hermite+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000629107000003}{[\mbox{Web of Science}]} $
• [44] H.W.J. Zhang, (q;c)-derivative operator and its applications, Adv. Appl. Math. 121 (2020), 102081, 23 pp. $ \href{https://doi.org/10.1016/j.aam.2020.102081}{[\mbox{CrossRef}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000577542300003}{[\mbox{Web of Science}]} $
• [45] S. Arjika and M.K. Mahaman, q-difference equation for generalized trivariate q-Hahn polynomials, Appl. Anal. Optim. 5(1) (2021), 1–11. $ \href{http://yokohamapublishers.jp/online2/opaao/vol5/p1.html}{[Web]} $
• [46] J. Cao, B.B. Xu and S. Arjika, A note on generalized q-difference equations for general Al-Salam-Carlitz polynomials, Adv. Differ. Equ.,2020(1) (2020), 668. $ \href{https://doi.org/10.1186/s13662-020-03133-7}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85095110366&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+note+on+generalized+%24q%24-difference+equations+for+general+Al-Salam-Carlitz+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000619928500002}{[\mbox{Web of Science}]} $
• [47] J. Cao, J.Y. Huang, M. Fadel and S. Arjika, A review of q-difference equations for Al-Salam-Carlitz polynomials and applications toU(n+1) type generating functions and Ramanujan’s integrals, Mathematics. 11(7) (2023), 1655. $ \href{https://doi.org/10.3390/math11071655}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85152798084&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+review+of+%24q%24-difference+equations+for+Al-Salam-Carlitz+polynomials+and+applications+to+%24U%28n%2B1%29%24+type+generating+functions+and+Ramanujan%27s+integrals%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000968779500001}{[\mbox{Web of Science}]} $
• [48] W. Hahn, Über Orthogonalpolynome, die q-Differenzengleichungen genu¨gen, Math. Nachr. 2(1-2) (1949), 4–34. $ \href{http://dx.doi.org/10.1002/mana.19490020103}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0003124908&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Uber+Orthogonalpolynome%2C+die+q-Differenzengleichungen+gen%C3%BCgen%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=1}{[\mbox{Scopus}]} $
• [49] R. Koekoek and R.F. Swarttouw, The Askey scheme of hypergeometric orthogonal polynomials and its q-analogue, Tech. Rep. 98–17, Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, 1998. $ \href{https://fa.ewi.tudelft.nl/~koekoek/askey/}{[Web]} $
• [50] R. Askey, Limits of some q-Laguerre polynomials, J. Approx. Theory. 46(3) (1986), 213–216. $ \href{https://doi.org/10.1016/0021-9045(86)90062-6}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0013380243&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Limits+of+some+%24q%24-Laguerre+polynomials%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1986C591100001}{[\mbox{Web of Science}]} $
• [51] B. Malgrange, Lectures on Functions of Several Complex Variables, Springer-Verlag, Berlin, 1984. $ \href{https://doi.org/10.1007/978-3-319-11511-5}{[\mbox{CrossRef}]} $
• [52] J.S. Christiansen, The moment problem associated with the q-Laguerre polynomials, Constr. Approx. 19(1)(2003), 1–22. $ \href{https://doi.org/10.1007/s00365-001-0017-5}{[\mbox{CrossRef}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000179482600001}{[\mbox{Web of Science}]} $
• [53] M.E.H. Ismail and M. Rahman, The q-Laguerre polynomials and related moment problems, J. Math. Anal. Appl. 218(1) (1998), 155–174. $ \href{https://doi.org/10.1006/jmaa.1997.5771}{[\mbox{CrossRef}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000072001000012}{[\mbox{Web of Science}]} $
• [54] D.S. Moak, The q-analogue of the Laguerre polynomials, J. Math. Anal. Appl., 81(1) (1981), 20–47. $ \href{https://doi.org/10.1016/0022-247X(81)90048-2}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0003009316&origin=resultslist&sort=plf-f&src=s&sid=c06d561f1acc2eac1727c85d5bdb36d3&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+%24q%24-analogue+of+the+Laguerre+polynomials%22%29&sl=103&sessionSearchId=c06d561f1acc2eac1727c85d5bdb36d3&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1981LU91300003}{[\mbox{Web of Science}]} $
• [55] J. Taylor, Several Complex Variables with Connections to Algebraic Qeometry and Lie Qroups, Graduate Studies in Mathematics, 46, American Mathematical Society, Providence, 2002.
• [56] G.E. Andrews and R. Askey, Enumeration of Partitions: The Role of Eulerian Series and q-Orthogonal Polynomials, in Higher Combinatories (M. Aigner, Ed.), 3–26, Reidel, Boston, MA (1977). $ \href{https://doi.org/10.1007/978-94-010-1220-1_1}{[\mbox{CrossRef}]} $
• [57] G.E. Andrews and R. Askey, Classical orthogonal polynomials, in: C. Brezinski et al., Eds., PolynBmes Orthogonaux et Applications, Lecture Notes in Math. 1171 (Springer, New York, 1985), 36–62. $ \href{https://doi.org/10.1007/BFb0076530}{[\mbox{CrossRef}]} $
• [58] G.E. Andrews, An introduction to Ramanujan’s ”lost” notebook, Amer. Math. Monthly 86(2) (1979), 89–108. $\href{https://doi.org/10.2307/2321943}{[\mbox{CrossRef}]} $
• [59] R. Askey, Two integrals of Ramanujan, Proc. Amer. Math. Soc. 85(2) (1982), 192–194. $ \href{https://doi.org/10.2307/2044279}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84966238721&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Two+integrals+of+Ramanujan%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=1}{[\mbox{Scopus}]} $
• [60] G.E. Andrews and R. Askey, Another q-extension of the beta function, Proc. Amer. Math. Soc. 81(1) (1981), 97–100. $ \href{https://doi.org/10.2307/2043995}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84966253958&origin=resultslist&sort=plf-f&src=s&sid=17f01c644f2f930366c8206c1af8bd5c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Another+%24q%24-extension+of+the+beta+function%22%29&sl=46&sessionSearchId=17f01c644f2f930366c8206c1af8bd5c&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1981KY78700021}{[\mbox{Web of Science}]} $
• [61] V.Y.B. Chen and N.S.S. Gu, The Cauchy operator for basic hypergeometric series, Adv. Appl. Math. 41(2) (2008), 177–196. $ \href{https://doi.org/10.1016/j.aam.2007.08.001}{[\mbox{CrossRef}]} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-43449110068&origin=resultslist&sort=plf-f&src=s&sid=091c2ace3e7c91bf2f9e1975aebaa5be&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+Cauchy+operator+for+basic+hypergeometric+series%22%29&sl=66&sessionSearchId=091c2ace3e7c91bf2f9e1975aebaa5be&relpos=0}{[\mbox{Scopus}]} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000257149000003}{[\mbox{Web of Science}]} $