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: Open access

PDF File (170 KB)

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.
    Mathematical Reviews (MathSciNet): MR2049636
    Digital Object Identifier: doi:10.4134/CKMS.2003.18.4.781
  • J. Choi, A generalization of Gottlieb polynomials in several variables, Appl. Math. Lett. 25 (2012), 43–46.
    Mathematical Reviews (MathSciNet): MR2826647
    Digital Object Identifier: doi:10.1016/j.aml.2011.07.006
  • J. Choi, $q$-Extension of a generalization of Gottlieb polynomials in two variables, J. Chungcheong Math. Soc. 25 (2012), 253–265.
    Mathematical Reviews (MathSciNet): MR3013452
    Digital Object Identifier: doi:10.5831/HMJ.2012.34.3.327
  • J. Choi, $q$-Extension of a generalization of Gottlieb polynomials in three variables, Honam Math. J. 34 (2012), 327–340.
    Mathematical Reviews (MathSciNet): MR3013452
    Digital Object Identifier: doi:10.5831/HMJ.2012.34.3.327
  • 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.
    Mathematical Reviews (MathSciNet): MR1052153
  • M. J. Gottlieb, Concerning some polynomials orthogonal on a finite or enumerable set of points, Amer. J. Math. 60(2) (1938), 453–458.
    Mathematical Reviews (MathSciNet): MR1507326
    Digital Object Identifier: doi:10.2307/2371307
  • 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.
    Mathematical Reviews (MathSciNet): MR2919021
    Digital Object Identifier: doi:10.4134/CKMS.2012.27.1.159
  • 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.
    Mathematical Reviews (MathSciNet): MR2776557
  • 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.
    Mathematical Reviews (MathSciNet): MR279355
  • J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Soc. 9 (1934), 6–13.
    Mathematical Reviews (MathSciNet): MR1574715
    Digital Object Identifier: doi:10.1112/jlms/s1-9.1.6
  • E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
    Mathematical Reviews (MathSciNet): MR393590
  • 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.
    Mathematical Reviews (MathSciNet): MR577575
  • 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.
    Mathematical Reviews (MathSciNet): MR577575
  • H. M. Srivastava, Certain $q$-polynomial expansions for functions of several variables, IMA J. Appl. Math. 30 (1983), 315–323.
    Mathematical Reviews (MathSciNet): MR719983
    Digital Object Identifier: doi:10.1093/imamat/30.3.315
  • H. M. Srivastava, Certain $q$-polynomial expansions for functions of several variables. II, IMA J. Appl. Math. 33 (1984), 205–209.
    Mathematical Reviews (MathSciNet): MR767521
    Digital Object Identifier: doi:10.1093/imamat/33.2.205
  • H. M. Srivastava, A family of $q$-generating functions, Bull. Inst. Math. Acad. Sinica 12 (1984), 327–336.
    Mathematical Reviews (MathSciNet): MR794402
  • 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.
    Mathematical Reviews (MathSciNet): MR834385
  • 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.
    Mathematical Reviews (MathSciNet): MR750112
  • G. Szegö, Orthogonal Polynomials, Fourth edition, Amererican Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, Rhode Island, 1975.
    Mathematical Reviews (MathSciNet): MR372517

Publication Committee for the Tokyo Journal of Mathematics

Tokyo Journal of Mathematics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more