The Umbral operator and the integration involving generalized Bessel-type functions
Kottakkaran Sooppy Nisar ; Saiful Rahman Mondal ; Praveen Agarwal ; Mujahed Al-Dhaifallah
Open Mathematics, Tome 13 (2015), / Harvested from The Polish Digital Mathematics Library

The main purpose of this paper is to introduce a class of new integrals involving generalized Bessel functions and generalized Struve functions by using operational method and umbral formalization of Ramanujan master theorem. Their connections with trigonometric functions with several distinct complex arguments are also presented.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:271004
@article{bwmeta1.element.doi-10_1515_math-2015-0041,
     author = {Kottakkaran Sooppy Nisar and Saiful Rahman Mondal and Praveen Agarwal and Mujahed Al-Dhaifallah},
     title = {The Umbral operator and the integration involving generalized Bessel-type functions},
     journal = {Open Mathematics},
     volume = {13},
     year = {2015},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0041}
}
Kottakkaran Sooppy Nisar; Saiful Rahman Mondal; Praveen Agarwal; Mujahed Al-Dhaifallah. The Umbral operator and the integration involving generalized Bessel-type functions. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0041/

[1] T. Amdeberhan and V. H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, Ramanujan J. 18 (2009), no. 1, 91–102. [Crossref][WoS] | Zbl 1178.33002

[2] T. Amdeberhan,O. Espinosa, I. Gonzalez, M. Harrison, V. H. Moll and A. Straub, Ramanujan’s master theorem, Ramanujan J. 29 (2012), no. 1-3, 103–120. [Crossref][WoS] | Zbl 1258.33001

[3] P. E. Appell and J. Kampé de Férit, Fonctions Hypergeometriques et Polynomes d’Hermite, Gauthier-Villars, Paris, 1926.

[4] D. Babusci, G. Dattoli, G. H. E. Duchamp, Góska and K. A. Penson, Definite integrals and operational methods, Appl. Math. Comput. 219 (2012), no. 6, 3017–3021. [WoS] | Zbl 1309.44002

[5] D. Babusci and G. Dattoli, On Ramanujan Master Theorem, arXiv:1103.3947 [WoS] | Zbl 1280.33005

[math-ph].

[6] D. Babusci, G. Dattoli, B. Germano, M. R. Martinelli and P.E. Ricci, Integrals of Bessel functions, Appl. Math. Lett. 26 (2013), no. 3, 351–354. [Crossref] | Zbl 1257.33003

[7] B. C. Berndt, Ramanujan’s notebooks. Part I, Springer, New York, 1985. | Zbl 0555.10001

[8] Á. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73 (2008), no. 1-2, 155–178. | Zbl 1156.33302

[9] K. Górska, D. Babusci, G. Dattoli, G. H. E. Duchamp and K. A. Penson, The Ramanujan master theorem and its implications for special functions, Appl. Math. Comput. 218 (2012), no. 23, 11466–11471. [WoS] | Zbl 1280.33005

[10] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, fifth edition, Academic Press, San Diego, CA, 1996. | Zbl 0918.65001

[11] G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge Univ. Press, Cambridge, England, 1940. | Zbl 67.0002.09

[12] S. R. Mondal and A. Swaminathan, Geometric properties of generalized Bessel functions, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 1, 179–194. | Zbl 1232.30013

[13] R. Mullin and G.-C. Rota. On the foundations of combinatorial theory III. Theory of binomial enumeration. In B. Harris, editor, Graph theory and its applications, Academic Press, 1970, 167–213.

[14] G.-C. Rota. The number of partitions of a set. Amer. Math. Monthly, (1964),no. 71, 498–504. [Crossref] | Zbl 0121.01803

[15] G.-C. Rota. Finite operator calculus. Academic Press, New York, 1975.

[16] G.-C. Rota and B.D. Taylor. An introduction to the umbral calculus. In H.M. Srivastava and Th.M. Rassias, editors, Analysis, Geometry and Groups: A Riemann Legacy Volume, Palm Harbor, Hadronic Press, 1993, 513–525. | Zbl 0910.05010

[17] G.-C. Rota and B.D. Taylor. The classical umbral calculus. SIAM J. Math. Anal.,(1994),no 25, 694–711. | Zbl 0797.05006

[18] S. Roman, The Umbral Calculus (Academic Press, INC, Orlando, 1984). | Zbl 0536.33001

[19] H. M. Srivastava and H. L. Manocha, A treatise on generating functions, Ellis Horwood Series: Mathematics and its Applications, Horwood, Chichester, 1984. | Zbl 0535.33001

[20] F. G. Tricomi, Funzioni ipergeometriche confluenti (Italian), Ed. Cremonese, Roma, 1954.

[21] N. Yagmur and H. Orhan, Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal. 2013, Art. ID 954513, 6 pp. [WoS] | Zbl 1272.30033

[22] H. W. Gould and A. T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J. 29 (1962), 51–63. [Crossref] | Zbl 0108.06504

[23] G. Maroscia and P. E. Ricci, Hermite-Kampé de Fériet polynomials and solutions of boundary value problems in the half-space, J. Concr. Appl. Math. 3 (2005), no. 1, 9–29. | Zbl 1094.33008

[24] G. Dattoli 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 Cimento Soc. Ital. Fis. (4) 20 (1997), no. 2, 3–133.

[25] G. Dattoli, P. E. Ricci and C. Cesarano, Monumbral polynomials and the associated formalism, Integral Transforms Spec. Funct. 13 (2002), no. 2, 155–162. | Zbl 1030.33006

[26] G. Dattoli, P. E. Ricci and C. Cesarano, Beyond the monomiality: the monumbrality principle, J. Comput. Anal. Appl. 6 (2004), no. 1, 77–83. | Zbl 1096.33005

[27] C. Cesarano, D. Assante, A note on Generalized Bessel Functions, International Journal of Mathematical Models and Methods in Applied Sciences, vol. 7, p. 625–629

[28] G. Dattoli, C. Cesarano and D. Sacchetti, A note on truncated polynomials, Appl. Math. Comput. 134 (2003), no. 2-3, 595–605. [Crossref]