Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

A General Inversion Pair and ρ-deformation of Askey Scheme


Affiliations
1 Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Charotar University of Science and Technology, Changa-388 421, Dist: Anand, India
2 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, India
     

   Subscribe/Renew Journal


The present work incorporates the general inverse series relations involving p-Pochhammer symbol and p-Gamma function. A general class of ρ-polynomials is introduced by means of this general inverse pair which is used to derive the generating function relations and summation formulas for certain p-polynomials belonging to this general class. This includes the p-deformation of Jacobi polynomials, the Brafman polynomials and Konhauser polynomials. Moreover, the orthogonal polynomials of Racah and those of Wilson are also provided ρ-deformation by means of the general inversion pair. The generating function relations and summation formulas for these polynomials are also derived. We then emphasize on the combinatorial identities and obtain their ρ-deformed versions.

Keywords

ρ-Gamma Function, ρ-Pochhammer Symbol, ρ-Deformed Polynomials, Inverse Series Relation.
Subscription Login to verify subscription
User
Notifications
Font Size


  • Deligne, P., Etingof, P., Freed, D. S., Jeffrey, L. C., Kazhdan, D., Morgan, J. W., Morrison, D. R., Witten, E.: Quantum fields and strings:a course for mathematicians. American Mathematical Society (1999)
  • Diaz, R., Pariguan, E.: Quantum symmetric functions. Communications in Algebra 6(33), 1947–1978 (2005)
  • Diaz, R., Pariguan, E.: On Hypergeometric function and Pochhammer k-symbol. Divulgaciones Mathem ´aticas 15(2), 179–192 (2007)
  • Diaz, R., Teruel, C.: q,k-Generalized Gamma and Beta function. Journal of Nonlinear Mathematical Physics 12(1), 118–134 (2005)
  • Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge university press, Cambridge (1990)
  • Gehlot, K. S., Prajapati, J. C.: Fractional calculus of generalized k-Wright function. Journal of Fractional Calculus and Applications 4(2), 283–289 (2013)
  • Koekoek, R., Swarttouw, R. F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98-17. TU Delft University of Technology, The Netherlands (1998)
  • Rainville, E. D.: Special Functions. Chelsea Publishing Company, New York (1971)
  • Riordan, J.: Combinatorial Identities. John Willey and Sons. Inc., Chichester (1968)
  • Savalia, R. V., Dave, B. I.: p-Deformation of a general class of polynomials and its properties. Journal of the Indian Math. Soc. 85(1-2), 226–240 (2018)
  • Saxena, R. K., Daiya, J., Singh, A.: Integral transforms of the k-Generalized Mittag-Leffler function E γ,τ k,α,β (z). Le Matematiche LXIX(Fasc. II), 7–16 (2014)
  • Srivastava, H. M.: The Weyl fractional integral of a general class of polynomials. Boll. Un. Mat. Ital. 6(2B), 219–228 (1983)
  • Srivastava, H. M., Manocha, H. L.: A Treatise on Generating Function. Ellis Horwood Limited, John Willey and Sons, England (1984)
  • Wilson, J. A.: Hypergeometric series, recurrence relations and some new orthogonal polynomials. Thesis, University of Wisconsin, Madison (1978)
  • Wilson, J. A.: Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11(4), 690–701 (1980)

Abstract Views: 18

PDF Views: 0




  • A General Inversion Pair and ρ-deformation of Askey Scheme

Abstract Views: 18  |  PDF Views: 0

Authors

Rajesh V. Savalia
Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Charotar University of Science and Technology, Changa-388 421, Dist: Anand, India
B. I. Dave
Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, India

Abstract


The present work incorporates the general inverse series relations involving p-Pochhammer symbol and p-Gamma function. A general class of ρ-polynomials is introduced by means of this general inverse pair which is used to derive the generating function relations and summation formulas for certain p-polynomials belonging to this general class. This includes the p-deformation of Jacobi polynomials, the Brafman polynomials and Konhauser polynomials. Moreover, the orthogonal polynomials of Racah and those of Wilson are also provided ρ-deformation by means of the general inversion pair. The generating function relations and summation formulas for these polynomials are also derived. We then emphasize on the combinatorial identities and obtain their ρ-deformed versions.

Keywords


ρ-Gamma Function, ρ-Pochhammer Symbol, ρ-Deformed Polynomials, Inverse Series Relation.

References