q-Difference equations of moment integrals for transformational identities and generating functions

References

• W.A.Al-Salam, and L.Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30 (1965), pp. 47–61.
   Web of Science ®Google Scholar
• W.N.Bailey, On the analogue of Dixon's theorem for bilateral basic hypergeometric series, Quart. J. Math. Oxford 1 (1950), pp. 318–320.
   Google Scholar
• C.Berg, and M.E.H.Ismail, q-Hermite polynomials and classical orthogonal polynomials, Can. J. Math 48 (1996), pp. 43–63.
   Web of Science ®Google Scholar
• B.C.Berndt, Ramanujan's Notebooks. Part III, Springer, New York, NY, 1991.
   Google Scholar
• D.M.Bressoud, A simple proof of Mehler's formula for q-Hermite polynomials, Indiana Univ. Math. J. 29 (1980), pp. 577–580.
   Web of Science ®Google Scholar
• J.Cao, New proofs of generating functions for Rogers–Szegö polynomials, Appl. Math. Comput. 207 (2009), pp. 486–492.
   Web of Science ®Google Scholar
• J.Cao, Bivariate generating functions for Rogers–Szegö polynomials, Appl. Math. Comput. 217 (2010), pp. 2209–2216.
   Web of Science ®Google Scholar
• J.Cao, Alternative proofs of generating functions for Hahn polynomials and some applications, Infin. Dimen. Anal. Quant. Prob. Rel. Top. 14 (2011), pp. 571–590.
   Web of Science ®Google Scholar
• J.Cao, Moments for generating functions of Al-Salam–Carlitz polynomials, Abstr. Appl. Anal. (2012), 18 pp. doi: 10.1155/2012/548168.
   Web of Science ®Google Scholar
• L.Carlitz, Some polynomials related to theta functions, Ann. Mat. Pure Appl. 41 (1955), pp. 359–373.
   Google Scholar
• L.Carlitz, Generating functions for certain q-orthogonal polynomials, Collectanea Math. 23 (1972), pp. 91–104.
   Google Scholar
• W.Chu, and X.Wang, Telescopic creation for identities of Rogers–Ramanujan type, J. Differ. Equ. Appl. 18 (2012), pp. 167–183.
   Web of Science ®Google Scholar
• W.Chu, and W.L.Zhang, Bilateral Bailey lemma and Rogers–Ramanujan identities, Adv. Appl. Math. 42 (2009), pp. 358–391.
   Web of Science ®Google Scholar
• J.-P.Fang, q-differential operator identities and applications, J. Math. Anal. Appl. 332 (2007), pp. 1393–1407.
   Web of Science ®Google Scholar
• G.Gasper, and M.Rahman, Basic hypergeometric series, Cambridge University Press, Cambridge, MA, 1990.
   Google Scholar
• M.E.H.Ismail, A queueing model and a set of orthogonal polynomials, J. Math. Anal. Appl. 108 (1985), pp. 575–594.
   Web of Science ®Google Scholar
• M.E.H.Ismail, and M.Rahman, Some basic bilateral sums and integrals, Pac. J. Math. 170 (1995), pp. 497–515.
   Web of Science ®Google Scholar
• M.E.H.Ismail, and D.Stanton, Classical orthogonal polynomials as moments, Can. J. Math. 49 (1997), pp. 520–542.
   Web of Science ®Google Scholar
• M.E.H.Ismail, and D.Stanton, q-integral and moment representations for q-orthogonal polynomials, Can. J. Math. 54 (2002), pp. 709–735.
   Web of Science ®Google Scholar
• Z.-G.Liu, A new proof of the Nassrallah-Rahman integral, Act. Math. Sin. 41 (1998), pp. 405–410, (in Chinese).
   Google Scholar
• Z.-G.Liu, Some operator identities and q-series transformation formulas, Discrete 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
• D.-Q.Lu, q-difference equation and the Cauchy operator identities, J. Math. Anal. Appl. 359 (2009), pp. 265–274.
   Web of Science ®Google Scholar
• J.McLaughlin, A.V.Sills, and P.Zimmer, Rogers–Ramanujan–Slater type identities, Electron. J. Comb. 15 (2008), DS15, 59 pp.
   Google Scholar
• L.J.Rogers, On a three-fold symmetry in the elements of Heine's series, Proc. London Math. Soc. 24 (1893), pp. 171–179.
   Google Scholar
• H.Rosengren, A bilateral series involving basic hypergeometric functions. Theory and applications of special functions, Dev. Math. 13 (2005), pp. 361–366.
   Google Scholar
• H.M.Srivastava, and A.K.Agarwal, Generating functions for a class of q-polynomials, Ann. Mat. Pura Appl. (Ser. 4) 154 (1989), pp. 99–109.
   Web of Science ®Google Scholar
• H.M.Srivastava, and V.K.Jain, Some multilinear generating functions for q-Hermite polynomials, J. Math. Anal. Appl. 144 (1989), pp. 147–157.
   Web of Science ®Google Scholar
• H.M.Srivastava, and P.W.Karlsson, Multiple Gaussian hypergeometric series, Halsted Press (Ellis Horwood Limited, Chichester)/Wiley, New York/Chichester/Brisbane and Toronto, 1985.
   Google Scholar
• H.M.Srivastava, and H.L.Manocha, A treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester)/Wiley, New York/Chichester/Brisbane and Toronto, 1984.
   Google Scholar
• H.M.Srivastava, M.A.Pathan, and M.G.Bin-Saad, A certain class of generating functions involving bilateral series, ANZIAM. J. 44 (2003), pp. 475–483.
   Web of Science ®Google Scholar
• A.Verma, and V.K.Jain, Poisson kernel and multilinear generating functions of some orthogonal polynomials, J. Math. Anal. Appl. 146 (1990), pp. 333–352.
   Web of Science ®Google Scholar
• Z.-Z.Zhang, and J.Wang, Two operator identities and their applications to terminating basic hypergeometric series and q-integrals, J. Math. Anal. Appl. 312 (2005), pp. 653–665.
   Web of Science ®Google Scholar