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Dave, B. I.
- A New Class of Functions Suggested by the Generalized Basic Hypergeometric Function
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Authors
Affiliations
1 Department of Mathematics, The Maharaja Sayajirao University of Baroda,Vadodara-390 002, IN
2 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, IN
1 Department of Mathematics, The Maharaja Sayajirao University of Baroda,Vadodara-390 002, IN
2 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara-390 002, IN
Source
The Journal of the Indian Mathematical Society, Vol 84, No 3-4 (2017), Pagination: 161-181Abstract
We introduce an extended generalized basic hypergeometric function rΦs+p in which p tends to infinity together with the summation index. We define the difference operators and obtain infinite order difference equation, for which these new special functions are eigen functions. We derive some properties, as the order zero of this function, differential equation involving a particular hyper-Bessel type operators of infinite order, and contiguous function relations. A transformation formula and an l-analogue of the q-Maclaurin's series are also obtained.Keywords
Basic Hypergeometric Function, q-Derivative, q-Integral, Eigen Function, Infinite Order Difference Equation.
References
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- Meera H. Chudasama and B. I. Dave. A new class of functions suggested by the qhypergeometric function. The Mathematics Student, 85(3-4), (2016), 47-61.
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- p-Deformation of a General Class of Polynomials and its Properties
Abstract Views :240 |
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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.
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- H. M. Srivastava and H. L. Manocha. A treatise on generating function, Ellis horwood Limited, John Willey and Sons, 1984.
- q-Analogue of an Extended Jacobi Polynomial and its Properties
Abstract Views :170 |
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Authors
Affiliations
1 Department of Mathematics, The M.S. University of Baroda, Baroda-390 002, IN
1 Department of Mathematics, The M.S. University of Baroda, Baroda-390 002, IN
Source
The Journal of the Indian Mathematical Society, Vol 71, No 1-4 (2004), Pagination: 77-84Abstract
In this paper we define a q-analogue of the extended Jacobi polynomial H(α, β)n, l, m[(a):(b):(x)][Boll, Un. Mat. Ital., 6(2B) (1983), 219-228], and obtain its inverse series relation, two q-integral representations, and a q-difference equation.- Generalized Mittag-Leffler Matrix Function and Associated Matrix Polynomials
Abstract Views :446 |
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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, IN
2 Ramrao Adik Institute of Technology, Navi Mumbai, IN
3 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara, IN
1 Department of Mathematical Sciences, P. D. Patel Institute of Applied Sciences, Charotar University of Science and Technology, Changa, IN
2 Ramrao Adik Institute of Technology, Navi Mumbai, IN
3 Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara, IN
Source
The Journal of the Indian Mathematical Society, Vol 86, No 1-2 (2019), Pagination: 161-178Abstract
The Mittag-Leffler function has been found useful in solving certain problems in Science and Engineering. On the other hand, noticing the occurrence of certain matrix functions in Special functions’ theory in general and in Statistics and Lie group theory in particular, we introduce here a matrix analogue of a recently generalized form of Mittag-Leffler function. This function yields the matrix analogues of the Saxena-Nishimoto’s function, Bessel-Maitland function, Dotsenko function and the Elliptic Function. We obtain matrix differential equation and eigen matrix function property for the proposed matrix function. Also, a generalized Konhauser matrix polynomial is deduced and its inverse series relations and generating function are derived.Keywords
Mittag-Leffler Matrix Function, Matrix Differential Equation, Generalized Konhauser Matrix Polynomial, Generating Function.References
- Abul-Dahab, M. A., Bakhet A. K., A certain generalized gamma matrix functions and their properties, J. Ana. Num. Theor. 3(1) (2015), 63-68.
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- Jodar, L., Cortes, J. C., Some properties of Gamma and Beta matrix functions, Appl. Math. Lett., 11(1)(1998), 89-93
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- Mittag-Leffler, G., Sur la nouvelle fonction eα(x), C. R. Acad. Sci., Paris, 137(1903), 554–558.
- Nathwani, B. V., Dave, B. I., Generalized Mittag-Leffler function and its properties, The Mathemaics Student, 86(1-2)(2017), 63–76.
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- Saxena, R. K., Nishimoto, K. N., Fractional calculus of generalized Mittag-Leffler functions , J. Frac. Calc., 37(2010), 43–52.
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- A General Inversion Pair and ρ-deformation of Askey Scheme
Abstract Views :335 |
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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)