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Savalia, Rajesh V.
- p-Deformation of a General Class of Polynomials and its Properties
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Authors
Affiliations
1 Charotar University of Science and Technology, Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, IN
2 The Maharaja Sayajirao University of Baroda, Department of Mathematics, Vadodara, IN
1 Charotar University of Science and Technology, Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, IN
2 The Maharaja Sayajirao University of Baroda, Department of Mathematics, Vadodara, IN
Source
The Journal of the Indian Mathematical Society, Vol 85, No 1-2 (2018), Pagination: 226-240Abstract
The work incorporates the extension of the Srivastava-Pathan’s generalized polynomial by means of p-generalized gamma function: Γp and Pochhammer p-symbol (x)n,p due to Rafael Dıaz and Eddy Pariguan [Divulgaciones Mathematicas Vol.15, No. 2(2007), pp. 179-192]. We establish the inverse series relation of this extended polynomial with the aid of general inversion theorem. We also obtain the generating function relations and the differential equation. Certain p-deformed combinatorial identities are illustrated in the last section.Keywords
General Class of p-Deformed Polynomials, p-Deformed Inverse Series Relation, p-Deformed Combinatorial Identities.
References
- Manisha Dalbhide. Generalization of certain ordinary and basic polynomials system and their properties, Ph. D. Thesis. The Maharaja Sayajirao University of Baroda, (2004), ISBN 978-1339-36594-7, ProQuestLLC, Ann Arbor, MI 48106, USA, 2015.
- Rafael Diaz and Eddy Pariguan. Quantum symmetric functions, Communications in Algebra, 33(6) (2005), 1947-1978.
- Rafael Diaz and Eddy Pariguan. On hypergeometric function and pochhammer k-symbol. Divulgaciones Mathematicas, 15(2) (2007), 179-192.
- Rafael Diaz and Carolina Teruel. q,k-generalized gamma and beta function. Journal of Nonlinear Mathematical Physics, 12(1) (2005), 118-134.
- D. Freed L. Jerey J. Morgan D. Morrison P. Deligne, P. Etingof and E. Witten. Quantum elds and strings:a course for mathematicians. American Mathematical Society, 1999.
- E. D. Rainville. Special funtions, The Macmillan Company, New York, 1960.
- John Riordan. Combinatorial Identities, John Willey and Sons. Inc., 1868.
- H. M. Srivastava. The weyl fractional integral of a general class of polynomials. Boll. Un. Mat. Ital., 6(2B) (1983), 219-228.
- H. M. Srivastava and H. L. Manocha. A treatise on generating function, Ellis horwood Limited, John Willey and Sons, 1984.
- A General Inversion Pair and ρ-deformation of Askey Scheme
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Authors
Affiliations
1 Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Charotar University of Science and Technology, Changa-388 421, Dist: Anand, IN
2 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, IN
1 Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Charotar University of Science and Technology, Changa-388 421, Dist: Anand, IN
2 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, IN
Source
The Journal of the Indian Mathematical Society, Vol 86, No 3-4 (2019), Pagination: 296-314Abstract
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
- 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)