Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Integral and series representations of $ q$-polynomials and functions: Part II Schur polynomials and the Rogers-Ramanujan identities


Authors: Mourad E. H. Ismail and Ruiming Zhang
Journal: Proc. Amer. Math. Soc. 145 (2017), 3717-3733
MSC (2010): Primary 11P84, 33D45; Secondary 05A17
DOI: https://doi.org/10.1090/proc/13535
Published electronically: May 24, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give several expansion and identities involving the Ramanujan function $ A_q$ and the Stieltjes-Wigert polynomials. Special values of our identities give $ m$-versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometric functions.


References [Enhancements On Off] (What's this?)

  • [1] George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
  • [2] G. E. Andrews, A polynomial identity which implies the Rogers-Ramanujan identities, Scripta Math. 28 (1970), 297-305.
  • [3] Leonard Carlitz, A note on the Rogers-Ramanujan identities, Math. Nachr. 17 (1958), 23-26. MR 0095362
  • [4] Kristina Garrett, Mourad E. H. Ismail, and Dennis Stanton, Variants of the Rogers- Ramanujan identities, Adv. in Appl. Math. 23 (1999), no. 3, 274-299. MR 1722235, https://doi.org/10.1006/ aama.1999.0658
  • [5] George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., with a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. MR 2128719
  • [6] Mourad E. H. Ismail, The zeros of basic Bessel functions, the functions $ J_{\nu +ax}(x)$, and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), no. 1, 1-19. MR 649849, https://doi.org/10.1016/0022-247X(82)90248-7
  • [7] Mourad E. H. Ismail, Asymptotics of $ q$-orthogonal polynomials and a $ q$-Airy function, Int. Math. Res. Not. 18 (2005), 1063-1088. MR 2149641, https://doi.org/10.1155/IMRN.2005.1063
  • [8] Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, with two chapters by Walter Van Assche, with a foreword by Richard A. Askey. Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. Reprint of the 2005 original. MR 2542683
  • [9] Mourad E. H. Ismail and Ruiming Zhang, Diagonalization of certain integral operators, Adv. Math. 109 (1994), no. 1, 1-33. MR 1302754, https://doi.org/10.1006/aima.1994.1077
  • [10] Mourad E. H. Ismail and Ruiming Zhang, Chaotic and periodic asymptotics for $ q$-orthogonal polynomials, Int. Math. Res. Not. 2006, Art. ID 83274, 33 pp.. MR 2272095, https://doi.org/10.1155/IMRN/2006/83274
  • [11] M. E. H. Ismail and R. Zhang, $ q$-Bessel functions and Rogers-Ramanujan type identities, Proc. Amer. Math. Soc. (2017), to appear.
  • [12] M. E. H. Ismail and R. Zhang, Integral and series representations of $ q$-polynomials and functions: part I, to appear.
  • [13] Hans Rademacher, Topics in analytic number theory, edited by E. Grosswald, J. Lehner and M. Newman, Die Grundlehren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York-Heidelberg, 1973. MR 0364103
  • [14] I. Schur, ``Ein Beitrag zur Additiven Zahlentheorie und zur Theorie der Kettenbrüche'', Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalish??mathematische Klasse, 302-321, 1917; reprinted in I. Schur, Gesmmelte Abhandungen, Vol. 2, pp. 117-136, Springer-Verlag, Berlin, 1973. MR0462892
  • [15] L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147-167. MR 0049225
  • [16] E. T. Whittaker and G. N. Watson, A course of modern analysis, An introduction to the general theory of infinite processes and of analytic functions, with an account of the principal transcendental functions. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. Reprint of the fourth (1927) edition. MR 1424469
  • [17] Ruiming Zhang, Scaled asymptotics for some $ q$-series, Q. J. Math. 59 (2008), no. 3, 389-407. MR 2444069, https://doi.org/10.1093/qmath/ham045

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11P84, 33D45, 05A17

Retrieve articles in all journals with MSC (2010): 11P84, 33D45, 05A17


Additional Information

Mourad E. H. Ismail
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: mourad.eh.ismail@gmail.com

Ruiming Zhang
Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China
Email: ruimingzhang@yahoo.com

DOI: https://doi.org/10.1090/proc/13535
Keywords: Rogers-Ramanujan identities, $m$-versions, the Ramanujan function, Stieltjes--Wigert polynomials, bilateral $q$-series
Received by editor(s): May 9, 2016
Received by editor(s) in revised form: May 11, 2016, June 9, 2016, and October 11, 2016
Published electronically: May 24, 2017
Additional Notes: Research partially supported by the DSFP of King Saud University and by the National Plan for Science, Technology and innovation (MAARIFAH), King Abdelaziz City for Science and Technology, Kingdom of Saudi Arabia, Award number 14-MAT623-02
The second author is the corresponding author. His research was partially supported by the National Science Foundation of China, grant No. 11371294
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society