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http://dx.doi.org/10.5831/HMJ.2021.43.4.597

FINITE INTEGRALS ASSOCIATED WITH THE PRODUCT OF ORTHOGONAL POLYNOMIALS AND WRIGHT FUNCTION  

Khan, Nabiullah (Department of Applied Mathematics, Aligarh Muslim University)
Khan, Mohammad Iqbal (Department of Applied Mathematics, Aligarh Muslim University)
Khan, Owais (Department of Mathematics, Integral University)
Publication Information
Honam Mathematical Journal / v.43, no.4, 2021 , pp. 597-612 More about this Journal
Abstract
Several useful and interesting extensions of the various special functions have been introduced by many authors during the last few decades. Various integral formulas associated with Wright function have been studied and a noteworthy amount of work have found in literature. The principal object of the present paper is to evaluate finite integral formulas containing the product of orthogonal polynomials with generalized Wright function. These integral formulas are expressed in terms of Srivastava and Daoust function. Some interesting particular cases are obtained from the main results by specialising the suitable values of the parameters involved.
Keywords
generalized Wright function; generalized hypergeometric function; Prudnikov integral; orthogonal polynomials;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 N.U. Khan, M.Ghayasuddin and T. Usman ,On certain integral formulas involving the product of Bessel function and Jacobi polynomial, Tamkang Journal of Mathematics, 47(3) (2016), 151-153.
2 A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integral and Series V.3.More special functions, New York-London: Gordon and Breach, (1992).
3 I. Podlubny, Fractional differential equations, Academic Press, New York, USA, (1999).
4 M. El-Shahed and A. Salem, An extension of Wright function and its properties, J. Mathematics, 2015, (2015) Article ID 950728, 1-11.
5 H.M. Srivastava and H.L. Manocha, A Treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and sons, New York, (1984).
6 H.M. Srivastava and M.C. Daoust, Certain generalized Neumann expansions associated with the kampe de Feriet function, Nederl. Akad. Wetensh. Proc. Ser. A Indag Math. 31 (1969), 449-457.
7 H.M. Srivastava and M.C. Daoust, A note on the convergence of Kampe de Feriet's double hypergeometric series, Math.Nachr. 53 (1972), 151-159.   DOI
8 D.L. Suthar, S.D. Purohit, H. Habenom and J. Singh, Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function, Discrete and Continous Dynamical Systems Series S. (2021), doi: 10.3934/dcdss.2021019.   DOI
9 E.T. Whittaker and G.N. Watson, A course of modern analysis, 4th edition, Cambridge University Press, Cambridge London, New York, (1927).
10 A. Wiman, Ber den Fundamentalsatz in der Theoric der Funktionen Eα(x), Acta Math. 29 (1905), 191-201.   DOI
11 D.L. Suthar, K. Tilahun and A. Oli, Generalized Fractional Calculus Operators Involving the Product of the Jacobi Type Orthogonal Polynomials and Multivariable Polynomials, Applications and Applied Mathematics, 15(1) (2020), 565-581.
12 N.U. Khan, T. Usman and M. Aman, Generalized Wright function and its properties using extended beta function, Tamkang journal of Mathematics, 51(4) (2020), 349-363.   DOI
13 L.C. Andrews, Special functions for engineers and applied mathematicians, MacMillan, New York, USA, 1985.
14 C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc. 27 (1928), 389-400.   DOI
15 N.U. Khan, T. Usman and M.Ghayasuddin, A unified double integral associated with Whittaker functions, Journal of Nonlinear Systems and Applications, 5 (2016), 21-24.
16 N.U. Khan, M. Ghayasuddin, W. A. Khan and Sarvat Zia, On integral operator involving Mittag-Leffler function, J. Ramanujan Soc. Math. and Sci. 5(1) (2016), 147-154.
17 N.U. Khan, Raghib Nadeem, T. Usman and A.H. Khan, Evaluation of integral formulas associated with the product of generalized Bessel function with orhogonal polynomials, Homan Mathematical J. 41 (2019), 135-152.
18 N.U. Khan, T. Usman and M. Aman, Some properties concerning the analysis of generalized Wright function, Journal of Computational and Applied Mathematics, 376 (2020) 112840.   DOI
19 V. Kiryakova, Some special functions related to fractional calculus and fractional (non integer) order control systems and equations, Facta Universitatis Series: Automatic Control and Robotics, 7 (1) (2008), 79-98.
20 G. M. Mittag-Leffler, Sur la nouvellefonction Eα(x), Comptes Rendus de I'Academie des Sciences Paris, 137 (1903), 554-558.
21 N.U. Khan, M. Aman and T. Usman, Extended Beta, Hypergeometric and Confluent Hypergeometric functions, Transactions Issue Mathematics Series of physical- technical and mathematics science, Azerbaijan National Academy of Science, 39(1) 83-97.
22 N.U. Khan, T. Usman, M. Aman, S. Al-Omari and S. Araci, Computations of certain integral formulas involving generalized Wright function, Advances in Difference Equations, 2020:491
23 J. Choi, N.U. Khan, T. Usman and M. Aman, Certain unified polynomials, Integral Transforms and Special Functions 30(1) (2019), 1-13.   DOI
24 J. Choi, P. Agarwal, S. Mathur and S.D. Purohit, Certain new integral formulas involving the generalized Bessel functions , Bull. Korean Math. Soc. 51 (4) (2014), 995-1003.   DOI
25 N.U. Khan, M.Ghayasuddin and S.Wali Khan, Some finite integrals involving the product of Bessel function with Jacobi and Laguerre polynomials, Commun. Korean Math. Soc. 33(3) (2018), 1013-1024.   DOI
26 K.S. Nisar, D.L. Suthar, S.D. Purohit and H. Amsalu, Unified Integrals Involving Product of Multivariable Polynomials and Generalized Bessel Functions, Bol. Soc. Paran. Mat. (3s.) 38(6) (2020), 73-83.
27 E.D. Rainville, Special functions, The Macmillan Company, New York (1960).
28 H.M. Srivastava and J. Choi, Zeta and q-Zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, (2012).
29 N.U. Khan, T. Usman and M. Ghayasuddin, Some integrals associated with multiindex Mittag-Leffler functions,J. Appl. Math. and Informatics, 34(3) (2016), 249-255.   DOI
30 M. Kamarujjama, O. Khan, Computation of new class of integrals involving generalized Galue type Struve function, J. Comput. Appl. Math. 351 (2019), 228-236.   DOI
31 N.U. Khan, M. Ghayasuddin, W. A. Khan and Sarvat Zia, Certain unified integral involving generalized Bessel-Maitland function,South East Asian J. of Math and Math. Sci. 11(2) (2015), 27-36.
32 J. Choi and P. Agarwal, Certain unified integrals involving a product of Bessel functions of first kind, Honam Mathematical J. 35 (4) (2013), 667-677.   DOI
33 D.L. Suthar, H. Amsalu and K. Godifey, Certain integrals involving multivariate Mittag-Leffler function, Journal of Inequalities and Applications, (2019) 2019:208.   DOI
34 H. Tadesse, D.L. Suthar, and M. Ayalew, Finite Integral Formulas Involving Multivariable Aleph-Functions, Journal of Applied Mathematics, 2019, Article ID 6821797, 10 pages.
35 O. Khan, M. Kamarujjama and N.U. Khan, Certain integral transforms involving the Product of Galue type Stuve function and Jacobi Polynomial, Palestine Journal of Mathematics, 8(2) (2019), 191-199.