# CERTAIN NEW INTEGRAL FORMULAS INVOLVING THE GENERALIZED BESSEL FUNCTIONS

• Choi, Junesang ;
• Agarwal, Praveen ;
• Mathur, Sudha ;
• Purohit, Sunil Dutt
• Published : 2014.07.31
• 59 12

#### Abstract

A remarkably large number of integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas involving various Bessel functions have been presented. Very recently, Choi and Agarwal derived two generalized integral formulas associated with the Bessel function $J_{\nu}(z)$ of the first kind, which are expressed in terms of the generalized (Wright) hypergeometric functions. In the present sequel to Choi and Agarwal's work, here, in this paper, we establish two new integral formulas involving the generalized Bessel functions, which are also expressed in terms of the generalized (Wright) hypergeometric functions. Some interesting special cases of our two main results are presented. We also point out that the results presented here, being of general character, are easily reducible to yield many diverse new and known integral formulas involving simpler functions.

#### Keywords

Gamma function;hypergeometric function $_pF_q$;generalized (Wright) hypergeometric functions $_p{\Psi}_q$;Bessel functions;generalized Bessel function of the first kind;Oberhettinger's integral formula

#### References

1. A. Baricz, Generalized Bessel Functions of the First Kind, Springer-Verlag, Berlin, Hei-delberg, 2010.
2. Y. A. Brychkov, Handbook of Special Functions, Derivatives, Integrals, Series and Other Formulas, CRC Press, Taylor & Francis Group, Boca Raton, London, and New York, 2008.
3. J. Choi and P. Agarwal, Certain unified integrals associated with Bessel functions, Bound. Value Probl. 2013 (2013), no. 95, 9 pp. https://doi.org/10.1186/1687-2770-2013-9
4. J. Choi, A. Hasanov, H. M. Srivastava, and M. Turaev, Integral representations for Srivastava's triple hypergeometric functions, Taiwanese J. Math. 15 (2011), no. 6, 2751-2762. https://doi.org/10.11650/twjm/1500406495
5. C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc. (2) 27 (1928), 389-400.
6. M. Garg and S. Mittal, On a new unified integral, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 2, 99-101. https://doi.org/10.1007/BF02829845
7. P. Malik, S. R. Mondal, and A. Swaminathan, Fractional integration of generalized Bessel function of the first kind, IDETC/CIE, 2011, USA.
8. F. Oberhettinger, Tables of Mellin Transforms, Springer-Verlag, New York, 1974.
9. 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.
10. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1985.
11. E. M. Wright, The asymptotic expansion of the generalized hypergeometric functions, J. London Math. Soc. 10 (1935), 286-293.
12. E. M. Wright, The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London A 238 (1940), 423-451. https://doi.org/10.1098/rsta.1940.0002
13. E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. 46 (1940), no. 2, 389-408.
14. A. Baricz, Geometric properties of generalized Bessel functions of complex order, Mathematica 48(71) (2006), no. 1, 13-18.
15. A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73 (2008), no. 1-2, 155-178.
16. A. Baricz, Jordan-type inequalities for generalized Bessel functions, J. Inequal. Pure Appl. Math. 9 (2008), no. 2, Art. 39, 6 pp.

#### Cited by

1. SOME INTEGRALS ASSOCIATED WITH MULTIINDEX MITTAG-LEFFLER FUNCTIONS vol.34, pp.3_4, 2016, https://doi.org/10.14317/jami.2016.249
2. Certain families of integral formulas involving Struve functions vol.37, pp.3, 2017, https://doi.org/10.5269/bspm.v37i3.31890
3. Inequalities on an extended Bessel function vol.2018, pp.1, 2018, https://doi.org/10.1186/s13660-018-1656-4

#### Acknowledgement

Supported by : National Research Foundation of Korea