Gould-Hopper matrix-Bessel and Gould-Hopper matrix-Tricomi functions and related integral representations

Ruma Qamar; Tabinda Nahid; Mumtaz Riyasat; Naresh Kumar; Anish Khan; Ruma Qamar; Tabinda Nahid; Mumtaz Riyasat; Naresh Kumar; Anish Khan

doi:10.3934/math.2020296

AIMS Mathematics

2020, Volume 5, Issue 5: 4613-4623. doi: 10.3934/math.2020296

Previous Article Next Article

Research article

Gould-Hopper matrix-Bessel and Gould-Hopper matrix-Tricomi functions and related integral representations

1.
Department of Mathematics, IFTM University, Moradabad, India
2.
Department of Mathematics, Aligarh Muslim University, Aligarh, India
3.
Department of Applied Mathematics, Faculty of Engineering, Aligarh Muslim University, Aligarh, India
4.
Chemistry Department, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia
5.
Center of Excellence for Advanced Materials Research, King Abdulaziz University, Jeddah, Saudi Arabia

Received: 07 February 2020 Accepted: 12 May 2020 Published: 25 May 2020
MSC : 26A33, 33B10, 33C45

The paper performs an investigation on the new class of functions, namely the GouldHopper-Bessel matrix functions and Gould-Hopper-Tricomi matrix functions via operational methods. The generating functions, operational representations and connection formulae are established. The generalized forms of the Gould-Hopper-Bessel matrix and Gould-Hopper-Tricomi matrix functions are introduced using integral transform. Several important properties related to these functions are also deduced.
- Gould-Hopper matrix polynomials,
- Bessel functions,
- Tricomi functions,
- Euler's integral,
- operational methods
Citation: Ruma Qamar, Tabinda Nahid, Mumtaz Riyasat, Naresh Kumar, Anish Khan. Gould-Hopper matrix-Bessel and Gould-Hopper matrix-Tricomi functions and related integral representations[J]. AIMS Mathematics, 2020, 5(5): 4613-4623. doi: 10.3934/math.2020296

Related Papers:

Abstract

The paper performs an investigation on the new class of functions, namely the GouldHopper-Bessel matrix functions and Gould-Hopper-Tricomi matrix functions via operational methods. The generating functions, operational representations and connection formulae are established. The generalized forms of the Gould-Hopper-Bessel matrix and Gould-Hopper-Tricomi matrix functions are introduced using integral transform. Several important properties related to these functions are also deduced.

References

[1]	M. Ali, T. Nahid, S. Khan, Some results on hybrid relatives of the Sheffer polynomials via operational rules, Miskolc Math. Notes, 20 (2019), 729-743.
[2]	B. Çekim, R. Aktaş, Multivariable matrix generalization of Gould-Hopper polynomials, Miskolc Math. Notes, 16 (2015), 79-89. doi: 10.18514/MMN.2015.1112
[3]	G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle, 1999.
[4]	G. Dattoli, Generalized polynomials, operational identities and their applications, J. Comput. Appl. Math., 118 (2000), 111-123. doi: 10.1016/S0377-0427(00)00283-1
[5]	G. Dattoli, P. L. Ottaviani, A. Torre, et al. Evolution operator equations: Integration with algebraic and finite-difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cim., 20 (1997), 1-133.
[6]	G. Dattoli, P. E. Ricci, C. Cesarano, et al. Special polynomials and fractional calculus, Math. Comput. Model., 37 (2003), 729-733. doi: 10.1016/S0895-7177(03)00080-3
[7]	E. Defez, L. Jodar, Some applications of the Hermite matrix polynomials series expansions, J. Comput. Appl. Math., 99 (1998), 105-117. doi: 10.1016/S0377-0427(98)00149-6
[8]	A. J. Durán, Markovs theorem for orthogonal matrix polynomials, Can. J. Math., 48 (1996), 1180-1195. doi: 10.4153/CJM-1996-062-4
[9]	A. J. Durán, W. Van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl., 219 (1995), 261-280. doi: 10.1016/0024-3795(93)00218-O
[10]	J. S. Geronimo, Matrix orthogonal polynomials in the unit circle, J. Math. Phys., 22 (1981), 1359-1365. doi: 10.1063/1.525073
[11]	J. S. Geronimo, Scattering theory and matrix orthogonal polynomials on the real line, Circ. Syst. Signal Pr., 1 (1982), 471-495. doi: 10.1007/BF01599024
[12]	W. Groenevelt, E. Koelink, A hypergeometric function transform and matrix valued orthogonal polynomials, Constr. Approx., 38 (2013), 277-309. doi: 10.1007/s00365-013-9207-1
[13]	L. Jódar, R. Company, Hermite matrix polynomials and second order matrix differential equations, Approximation Theory and its Applications, 12 (1996), 20-30.
[14]	L. Jódar, R. Company, E. Navarro, Laguerre matrix polynomials and systems of second order differential equations, Appl. Numer. Math., 15 (1994), 53-63. doi: 10.1016/0168-9274(94)00012-3
[15]	L. Jódar, R. Company, E. Ponsoda, Orthogonal matrix polynomials and systems of second order differential equations, Diff. Equ. Dynam. Syst., 3(3) (1995), 269-288.
[16]	E. D. Rainville, Special functions, Bronx, New York, 1971.
[17]	H. M. Srivastava, H. L. Manocha, A Treatise on Generating Functions, 1984.
[18]	J. F. Steffensen, The poweriod, an extension of the mathematical notion of power, Acta Math., 73 (1941), 333-366. doi: 10.1007/BF02392231
[19]	G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, 1966.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)