q-Difference equations for the generalized Cigler’s polynomials

References

• W.A. Al-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30 (1965), pp. 47–61.
   Web of Science ®Google Scholar
• G.E. Andrews, Summations and transformations for basic Appell series, J. London Math. Soc. 4 (1972), pp. 618–622.
   Web of Science ®Google Scholar
• G.E. Andrews, Applications of basic hypergeometric series, SIAM Rev. 16 (1974), pp. 441–484.
   Web of Science ®Google Scholar
• G.E. Andrews and R. Askey, Another q-extension of the beta function, Proc. Am. Math. Soc. 81 (1981), pp. 97–100.
   Web of Science ®Google Scholar
• R. Askey and R. Roy, More q-beta integral, Rocky Mountain J. Math. 16 (1986), pp. 365–372.
   Web of Science ®Google Scholar
• J. Cao, Notes on Askey--Roy integral and certain generating functions for q-polynomials, J. Math. Anal. Appl. 409 (2014), pp. 435–445.
   Web of Science ®Google Scholar
• J. Cao, Generalization of certain Carlitz’s trilinear and Srivastava-Agarwal type generating functions, J. Math. Anal. Appl. 412 (2014), pp. 841–851.
   Web of Science ®Google Scholar
• J. Cao, q-Difference equations for generalized homogeneous q-operators and certain generating functions, J. Differ. Equ. Appl. 20 (2014), pp. 837–851.
   Web of Science ®Google Scholar
• J. Cao, A note on generalized q-difference equations for q-beta and Andrews--Askey integral, J. Math. Anal. Appl. 412 (2014), pp. 841–851.
   Web of Science ®Google Scholar
• J. Cao, A note on moment integrals and some applications, J. Math. Anal. Appl. 410 (2014), pp. 348–360.
   Web of Science ®Google Scholar
• J. Cao, Homogeneous q-difference equations and generating functions for q-hypergeometric polynomials, Ramanujan. J. 40 (2016), pp. 177–192.
   Web of Science ®Google Scholar
• J. Cao, Homogeneous q-partial difference equations and some applications, Adv. Appl. Math. 84 (2017), pp. 47–72.
   Web of Science ®Google Scholar
• J. Cao and D.-W. Niu, A note on q-difference equations for Cigler’s polynomials, J. Differ. Equ. Appl. 22 (2016), pp. 1880–1892.
   Web of Science ®Google Scholar
• L. Carlitz, Generating functions for certain q-orthogonal polynomials, Collectanea Math. 23 (1972), pp. 91–104.
   Google Scholar
• W.Y.C. Chen, A.M. Fu, and B. Zhang, The homogeneous q-difference operator, Adv. Appl. Math. 31 (2003), pp. 659–668.
   Web of Science ®Google Scholar
• J.S. Christiansen, The moment problem associated with the q-Laguerre polynomials, Constr. Approx. 19 (2003), pp. 1–22.
   Web of Science ®Google Scholar
• J. Cigler, Operator methods for q-identities, Monatsh. Math. 88 (1979), pp. 87–105.
   Web of Science ®Google Scholar
• J. Cigler, Operator methods for q-identities II: q-Laguerre polynomials, Monatsh. Math. 91 (1981), pp. 105–117.
   Web of Science ®Google Scholar
• J.-P. Fang, Note on a q-contour intergal formula, Appl. Math. Comput. 233 (2014), pp. 292–297.
   Web of Science ®Google Scholar
• J.-P. Fang, Applications of a generalized q-difference equation, Adv. Differ. Equ. 267 (2014), p. 19.
   Google Scholar
• J.-P. Fang, q-difference equation and q-polynomials, Appl. Math. Comput. 248 (2014), pp. 550–561.
   Web of Science ®Google Scholar
• J.-P. Fang, Remarks on generalized q-difference equation, J. Differ. Equ. Appl. 21 (2015), pp. 934–953.
   Web of Science ®Google Scholar
• G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, 2nd ed., Vol. 96, Cambridge University Press, Cambridge, 2004.
   Google Scholar
• F.H. Jackson, On q-difinite integrals, Q. J. Pure. Appl. Math. 50 (1910), pp. 101–112.
   Google Scholar
• 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.
   Google Scholar
• Z.-G. Liu, Some operator identities and q-series traansformation formulas, Discret. Math. 265 (2003), pp. 119–139.
   Web of Science ®Google Scholar
• Z.-G. Liu, Two q-difference equations and q-operator identities, J. Differ. Equ. Appl. 16 (2010), pp. 1293–1307.
   Web of Science ®Google Scholar
• Z.-G. Liu, An extension of the non-terminating 6φ5 summation and the Askey-Wilson polynomials, J. Differ. Equ. Appl. 17 (2011), pp. 1401–1411.
   Web of Science ®Google Scholar
• Z.-G. Liu, A q-extension of a partial differential equation and the Hahn polynomials, Ramanujan J. 38 (2015), pp. 481–501.
   Web of Science ®Google Scholar
• Z.-G. Liu and J. Zeng, Two expansion formulas involving the Rogers-Szeg\"{o} polynomials with applications, Int. J. Number Theory 11 (2015), pp. 507–525.
   Web of Science ®Google Scholar
• D.-Q. Lu, q-Difference equation and the Cauchy operator identites, J. Math. Anal. Appl. 359 (2009), pp. 265–274.
   Web of Science ®Google Scholar
• S.C. Milne, Balanced 3φ2 summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997), pp. 93–187.
   Web of Science ®Google Scholar
• R.M. Range, Complex analysis: A brief tour into higher dimensions, Amer. Math. Monthly 110 (2003), pp. 89–108.
   Web of Science ®Google Scholar
• H.L. Saad and A.A. Sukhi, Another homogeneous q-difference operator, Appl. Math. Comput. 215 (2010), pp. 4332–4339.
   Web of Science ®Google Scholar
• H.M. Srivastava and A.K. Agarwal, Generating functions for a class of q-polynomials, Ann. Mat. Pura. Appl. 154 (1989), pp. 99–109.
   Web of Science ®Google Scholar
• M. Wang, A remark on Andrews-Askey integral, J. Math. Anal. Appl. 341 (2008), pp. 1487–1494.
   Web of Science ®Google Scholar
• M. Wang, Generalizations of Milne’s U(n+1) q-binomial theorem, Comput. Math. Appl. 58 (2009), pp. 80–87.
   Web of Science ®Google Scholar
• M. Wang, q-Integral representation of the Al-Salam-Carlitz polynomials, Appl. Math. Lett. 22 (2009), pp. 943–945.
   Web of Science ®Google Scholar
• M. Wang, A new probability distribution with applications, Pac. J. Math. 247 (2010), pp. 241–255.
   Web of Science ®Google Scholar
• H.S. Wilf, Generatingfunctionology, 2nd ed., Academic Press, Boston, MA, 1994.
   Google Scholar
• Z.-Z. Zhang and M. Liu, Applications of operator identities to the multiple q-binomial theorem and q-Gauss summation theorem, Discrete Math. 306 (2006), pp. 1424–1437.
   Web of Science ®Google Scholar