Article
MSC:
05A30,
11B65,
33D15,
33D45,
33D50,
35C11 | MR 3989274 | Zbl 07088812 | DOI: 10.21136/CMJ.2018.0470-17
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
$q$-partial difference equation; homogeneous generalized Al-Salam-Carlitz polynomial; generating function; Andrews-Askey integral; Ramanujan $q$-beta integral
$q$-partial difference equation; homogeneous generalized Al-Salam-Carlitz polynomial; generating function; Andrews-Askey integral; Ramanujan $q$-beta integral
Summary:
We study the condition on expanding an analytic several variables function in terms of products of the homogeneous generalized Al-Salam-Carlitz polynomials. As applications, we deduce bilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. We also gain multilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. Moreover, we obtain generalizations of Andrews-Askey integrals and Ramanujan $q$-beta integrals. At last, we derive $U(n+1)$ type generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials.
References:
[1] Abramov, S. A., Bronstein, M., Petkovšek, M.: On polynomial solutions of linear operator equations. Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, ISSAC'95 A. H. M. Levelt ACM Press, New York (1995), 290-296. DOI 10.1145/220346.220384 | MR 2280523 | Zbl 0914.65132
[2] Al-Salam, W. A., Carlitz, L.: Some orthogonal $q$-polynomials. Math. Nachr. 30 (1965), 47-61. DOI 10.1002/mana.19650300105 | MR 0197804 | Zbl 0135.27802
[3] Andrews, G. E.: Summations and transformations for basic Appell series. J. Lond. Math. Soc., II. Ser. 4 (1972), 618-622. DOI 10.1112/jlms/s2-4.4.618 | MR 0306559 | Zbl 0235.33003
[4] Andrews, G. E.: Applications of basic hypergeometric functions. SIAM Rev. 16 (1974), 441-484. DOI 10.1137/1016081 | MR 0352557 | Zbl 0299.33004
[5] Andrews, G. E., Askey, R.: Another $q$-extension of the beta function. Proc. Am. Math. Soc. 81 (1981), 97-100. DOI 10.2307/2043995 | MR 0589145 | Zbl 0471.33001
[6] Askey, R.: Two integrals of Ramanujan. Proc. Am. Math. Soc. 85 (1982), 192-194. DOI 10.2307/2044279 | MR 0652440 | Zbl 0503.33001
[7] Atakishiyev, N. M., Feinsilver, P.: Two Ramanujan's integrals with a complex parameter. Proceedings of the IV Wigner Symposium 1995 N. M. Atakishiyev et al. World Scientific, Singapore (1996), 406-412. DOI 10.1142/9789814531207 | MR 1474783 | Zbl 0948.33500
[8] Cao, J.: A note on $q$-integrals and certain generating functions. Stud. Appl. Math. 131 (2013), 105-118. DOI 10.1111/sapm.12002 | MR 3084689 | Zbl 1302.33015
[9] Cao, J.: A note on generalized $q$-difference equations for $q$-beta and Andrews-Askey integral. J. Math. Anal. Appl. 412 (2014), 841-851. DOI 10.1016/j.jmaa.2013.11.027 | MR 3147253 | Zbl 1308.39010
[10] Cao, J.: $q$-difference equations for generalized homogeneous $q$-operators and certain generating functions. J. Difference Equ. Appl. 20 (2014), 837-851. DOI 10.1080/10236198.2013.823955 | MR 3210317 | Zbl 1350.39004
[11] Cao, J.: Homogeneous $q$-difference equations and generating functions for $q$-hypergeometric polynomials. Ramanujan J. 40 (2016), 177-192. DOI 10.1007/s11139-015-9676-x | MR 3485998 | Zbl 1335.05023
[12] Carlitz, L.: Generating functions for certain $Q$-orthogonal polynomials. Collect. Math. 23 (1972), 91-104. MR 0316773 | Zbl 0273.33012
[13] Fang, J.-P.: $q$-difference equation and $q$-polynomials. Appl. Math. Comput. 248 (2014), 550-561. DOI 10.1016/j.amc.2014.10.010 | MR 3276709 | Zbl 1338.39016
[14] Fang, J.-P.: Remarks on a generalized $q$-difference equation. J. Difference Equ. Appl. 21 (2015), 934-953. DOI 10.1080/10236198.2015.1056176 | MR 3393878 | Zbl 1325.33012
[15] Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications 96, Cambridge University Press, Cambridge (2004). DOI 10.1017/CBO9780511526251 | MR 2128719 | Zbl 1129.33005
[16] Gunning, R. C.: Introduction to Holomorphic Functions of Several Variables Vol. I: Function theory. The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Florence (1990). DOI 10.1201/9780203750063 | MR 1052649 | Zbl 0699.32001
[17] Ismail, M. E. H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications 98, Cambridge University Press, Cambridge (2009). DOI 10.1017/CBO9781107325982 | MR 2542683 | Zbl 1172.42008
[18] Koekoek, R., Lesky, P. A., Swarttouw, R. F.: Hypergeometric Orthogonal Polynomials and Their $q$-Analogues. Springer Monographs in Mathematics, Springer, Berlin (2010). DOI 10.1007/978-3-642-05014-5 | MR 2656096 | Zbl 1200.33012
[19] Koekoek, R., Swarttouw, R. F.: 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).
[20] Liu, Z.-G.: Two $q$-difference equations and $q$-operator identities. J. Difference Equ. Appl. 16 (2010), 1293-1307. DOI 10.1080/10236190902810385 | MR 2738950 | Zbl 1223.39005
[21] 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 10.1080/10236190903530735 | MR 2836870 | Zbl 1229.05043
[22] Liu, Z.-G.: On the $q$-partial differential equations and $q$-series. The Legacy of Srinivasa Ramanujan B. C. Berndt Ramanujan Mathematical Society Lecture Notes Series 20, Ramanujan Mathematical Society, Mysore (2013), 213-250. MR 3221313 | Zbl 1310.05024
[23] Liu, Z.-G.: A $q$-extension of a partial differential equation and the Hahn polynomials. Ramanujan J. 38 (2015), 481-501. DOI 10.1007/s11139-014-9632-1 | MR 3423009 | Zbl 1326.05016
[24] Liu, Z.-G., Zeng, J.: Two expansion formulas involving the Rogers-Szegő polynomials with applications. Int. J. Number Theory 11 (2015), 507-525. DOI 10.1142/S1793042115500268 | MR 3325432 | Zbl 1327.11073
[25] Malgrange, B.: Lectures on the Theory of Functions of Several Complex Variables. Lectures on Mathematics and Physics. Mathematics 13. Tata Institute of Fundamental Research, Bombay; Springer, Berlin (1984). MR 0742775 | Zbl 0561.32006
[26] Milne, S. C.: Balanced $_3\phi_2$ summation theorems for $U(n)$ basic hypergeometric series. Adv. Math. 131 (1997), 93-187. DOI 10.1006/aima.1997.1658 | MR 1475046 | Zbl 0886.33014
[27] Wang, M.: A remark on Andrews-Askey integral. J. Math. Anal. Appl. 341 (2008), 1487-1494. DOI 10.1016/j.jmaa.2007.11.011 | MR 2398544 | Zbl 1142.33006
[28] Wang, M.: A new probability distribution with applications. Pac. J. Math. 247 (2010), 241-255. DOI 10.2140/pjm.2010.247.241 | MR 2718213 | Zbl 1205.60037
[29] Wilf, H. S.: Generatingfunctionology. Academic Press, Boston (1994). DOI 10.1016/c2009-0-02369-1 | MR 1277813 | Zbl 0831.05001
[30] Zhang, Z., Liu, M.: Applications of operator identities to the multiple $q$-binomial theorem and $q$-Gauss summation theorem. Discrete Math. 306 (2006), 1424-1437. DOI 10.1016/j.disc.2006.01.025 | MR 2237725 | Zbl 1095.05002
Search
Partner of
