Ramanujan-type continuous measures for classical $q$-polynomials
Teoretičeskaâ i matematičeskaâ fizika, Tome 105 (1995) no. 3, pp. 383-392 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

It is shown that Ramanujan-type measures for a hierarchy of classical $q$-orthogonal polynomials can be systematically built from simple cases of the continuous $q$-Hermite and $q^{-1}$-Hermite polynomials by using the Berg–Ismail procedure of attaching generating functions to measures. The application of this technique leads also to the evaluation of Ramanujan-type integrals for the Al-Salam–Chihara polynomials both when $0 and $q>1$, as well as for the product of four particular nonterminating basic hypergeometric functions ${}_2\phi _1$.
@article{TMF_1995_105_3_a3,
     author = {N. M. Atakishiyev},
     title = {Ramanujan-type continuous measures for classical $q$-polynomials},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {383--392},
     year = {1995},
     volume = {105},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_105_3_a3/}
}
TY  - JOUR
AU  - N. M. Atakishiyev
TI  - Ramanujan-type continuous measures for classical $q$-polynomials
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1995
SP  - 383
EP  - 392
VL  - 105
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_1995_105_3_a3/
LA  - ru
ID  - TMF_1995_105_3_a3
ER  - 
%0 Journal Article
%A N. M. Atakishiyev
%T Ramanujan-type continuous measures for classical $q$-polynomials
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1995
%P 383-392
%V 105
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_1995_105_3_a3/
%G ru
%F TMF_1995_105_3_a3
N. M. Atakishiyev. Ramanujan-type continuous measures for classical $q$-polynomials. Teoretičeskaâ i matematičeskaâ fizika, Tome 105 (1995) no. 3, pp. 383-392. http://geodesic.mathdoc.fr/item/TMF_1995_105_3_a3/

[1] Askey R., Ismail M. E. H., “A generalization of ultraspherical polynomials.”, Studies in Pure Mathematics, ed. P. Erdös, Birkhäuser, Boston, Massachusetts, 1983, 55–78 | DOI | MR

[2] Askey R., Wilson J. A., Mem. Amer. Math. Soc., 54, no. 319, 1985, 1–55 | MR

[3] Gasper G., Rahman M., Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990 | MR | Zbl

[4] Atakishiyev N. M., Rahman M., Suslov S. K., “On the classical orthogonal polynomials”, Constructive Approximation (to appear) | MR

[5] Atakishiyev N. M., Nagiyev Sh. M., Theor. Math. Phys., 98:2 (1994), 162–166 | DOI | MR

[6] Atakishiyev N. M., Teor. i Matem. Fiz., 99:1 (1994), 155–159 | MR

[7] Atakishiyev N. M., Teor. i Matem. Fiz., 102:1 (1995), 32–39 | MR

[8] Atakishiyev N. M., Nagiyev Sh. M., J. Phys. A: Math. Gen., 27:17 (1994), L611–L615 | DOI | MR | Zbl

[9] Atakishiyev N. M., Journal of Group Theory in Physics, 2:2 (1994)

[10] Atakishiyev N. M., “A Ramanujan-type measure for the Al-Salam and Ismail biorthogonal rational functions.”, Proceedings of the Workshop on Symmetries and Integrability of Difference Equations (May 22–29, 1994, Estérel, Québec, Canada) | MR

[11] Whittaker E. T., Watson G. N., A Course of Modern Analysis, 4-th edition, Cambridge University Press, Cambridge, 1984 | MR

[12] Berndt B. C., CRM Proceedings and Lecture Notes, 1, 1993, 1–63 | DOI | MR | Zbl

[13] Szegő G., Collected Papers, V. 1, ed. R. Askey, Birkhäuser, Basel, 1982, 793–805 | MR

[14] Al-Salam W. A., Carlitz L., Boll. Unione Matem. Ital., 12:3 (1957), 414–417 | MR | Zbl

[15] Carlitz L., Publicationes Mathematicæ, 5:3–4 (1958), 222–228 | MR | Zbl

[16] Askey R., $q$-series and Partitions, IMA Volumes in Mathematics and Its Applications, ed. D. Stanton, Springer-Verlag, New York, 1989, 151–158 | DOI | MR

[17] Ismail M. E. H., Masson D. R., Trans. Amer. Math. Soc., 346:1 (1994), 63–116 | DOI | MR | Zbl

[18] Atakishiyev N. M., Frank A., Wolf K. B., J. Math. Phys., 35:7 (1994), 3253–3260 | DOI | MR | Zbl

[19] Berg C., Ismail M. E. H., $Q$-Hermite polynomials and classical orthogonal polynomials, submitted | MR

[20] Rogers L. J., Proc. London Math. Soc., 26 (1895), 15–32 | DOI | Zbl

[21] Bressoud D. M., Indiana Univ. Math. Journ., 29:4 (1980), 577–580 | DOI | MR | Zbl

[22] Al-Salam W. A., Chihara T. S., SIAM J. Math. Anal., 7:1 (1976), 16–28 | DOI | MR | Zbl

[23] Ismail M. E. H., Wilson J., J. Approx. Theory, 36:1 (1982), 43–54 | DOI | MR | Zbl

[24] Pastro P. I., J. Math. Anal. Appl., 112:2 (1985), 517–540 | DOI | MR | Zbl

[25] Al-Salam W. A., Ismail M. E. H., Proc. Amer. Math. Soc., 121:2 (1994), 553–561 | DOI | MR | Zbl

[26] Ismail M. E. H., Rahman M., Pacific Journal of Mathematics (to appear)