Tokyo Journal of Mathematics

$q$-Extension of a Multivariable and Multiparameter Generalization of the Gottlieb Polynomials in Several Variables

Junesang CHOI and H. M. SRIVASTAVA

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

While considering some families of polynomials which are orthogonal on a finite or enumerable set of points, Gottlieb was led in the year 1938 to what are now popularly known as the Gottlieb polynomials $\varphi_n(x;\lambda)$. This one-parameter family of polynomials has ever since then been cited widely and investigated extensively in several books, monographs and journal articles. In the present sequel to some of the aforementioned investigations, we introduce and systematically investigate a basic (or $q$-) extension of a multivariable and multiparameter generalization of the Gottlieb polynomials $\varphi_n(x;\lambda)$. We also establish a set of three new families of generating functions for the generalized $q$-Gottlieb polynomials defined here.

Article information

Source
Tokyo J. Math., Volume 37, Number 1 (2014), 111-125.

Dates
First available in Project Euclid: 28 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1406552433

Digital Object Identifier
doi:10.3836/tjm/1406552433

Mathematical Reviews number (MathSciNet)
MR3264516

Zentralblatt MATH identifier
1300.33008

Subjects
Primary: 33C65: Appell, Horn and Lauricella functions 33C99: None of the above, but in this section
Secondary: 33C05: Classical hypergeometric functions, $_2F_1$ 33C20: Generalized hypergeometric series, $_pF_q$

Citation

CHOI, Junesang; SRIVASTAVA, H. M. $q$-Extension of a Multivariable and Multiparameter Generalization of the Gottlieb Polynomials in Several Variables. Tokyo J. Math. 37 (2014), no. 1, 111--125. doi:10.3836/tjm/1406552433. https://projecteuclid.org/euclid.tjm/1406552433


Export citation

References

  • J. Choi, Notes on formal manipulations of double series, Commun. Korean Math. Soc. 18 (2003), 781–789.
  • J. Choi, A generalization of Gottlieb polynomials in several variables, Appl. Math. Lett. 25 (2012), 43–46.
  • J. Choi, $q$-Extension of a generalization of Gottlieb polynomials in two variables, J. Chungcheong Math. Soc. 25 (2012), 253–265.
  • J. Choi, $q$-Extension of a generalization of Gottlieb polynomials in three variables, Honam Math. J. 34 (2012), 327–340.
  • G. Gasper and M. Rahman, Basic Hypergeometric Series (with a Foreword by Richard Askey), Encyclopedia of Mathematics and Its Applications, Vol. 35, Cambridge University Press, Cambridge, New York, Port Chester, Melbourne and Sydney, 1990; Second edition, Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, London and New York, 2004.
  • M. J. Gottlieb, Concerning some polynomials orthogonal on a finite or enumerable set of points, Amer. J. Math. 60(2) (1938), 453–458.
  • M. A. Khan and M. Akhlaq, Some new generating functions for Gottlieb polynomials of several variables, Internat. Trans. Appl. Sci. 1 (2009), 567–570.
  • M. A. Khan and M. Asif, A note on generating functions of $q$-Gottlieb polynomials, Commun. Korean Math. Soc. 27 (2012), 159–166.
  • M. A. Khan, A. H. Khan and M. Singh, Integral representations for the product of Krawtchouk, Meixner, Charlier and Gottlieb polynomials, Internat. J. Math. Anal. (Ruse) 5 (2011), 199–206.
  • G. Lauricella, Sulle funzioni ipergeometriche a più variabili, Rend. Circ. Mat. Palermo 7 (1893), 111–158.
  • E. B. McBride, Obtaining Generating Functions, Springer Tracts in Natural Philosophy, Vol. 21, Springer-Verlag, New York, Heidelberg and Berlin, 1971.
  • J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Soc. 9 (1934), 6–13.
  • E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  • H. M. Srivastava, Some bilateral generating functions for a certain class of special functions. I, Nederl. Akad. Wetensch. Proc. Ser. A 83$=$ Indag. Math. 42 (1980), 221–233.
  • H. M. Srivastava, Some bilateral generating functions for a certain class of special functions. II, Nederl. Akad. Wetensch. Proc. Ser. A 83$=$ Indag. Math. 42 (1980), 234–246.
  • H. M. Srivastava, Certain $q$-polynomial expansions for functions of several variables, IMA J. Appl. Math. 30 (1983), 315–323.
  • H. M. Srivastava, Certain $q$-polynomial expansions for functions of several variables. II, IMA J. Appl. Math. 33 (1984), 205–209.
  • H. M. Srivastava, A family of $q$-generating functions, Bull. Inst. Math. Acad. Sinica 12 (1984), 327–336.
  • H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
  • H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
  • G. Szegö, Orthogonal Polynomials, Fourth edition, Amererican Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, Rhode Island, 1975.