Abstract
Inspired by certain recent extensions of the Euler’s beta, Gauß hypergeometric and confluent hypergeometric functions (Choi et al. in Honam Math 36(2):339–367, 2014), we introduce (p, q)-extended Bessel function \(J_{\nu ,p,q}\), the (p, q)-extended modified Bessel function \(I_{\nu ,p,q}\) of the first kind of order \(\nu \) by making use two additional parameters in the integrand, as well as the (p, q)-extended Struve \(\mathbf{H}_{\nu ,p,q}\) and the modified Struve \(\mathbf{L}_{\nu ,p,q}\) functions. Systematic investigation of its properties, among others integral representations, bounding inequalites Mellin transforms (for all newly defined Bessel and Struve functions), complete monotonicity, Turán type inequality, associated non-homogeneous differential-difference equations (exclusively for extended Bessel functions) are presented. Brief presentation of another members of Bessel functions family: spherical, ultraspherical, Delerue hyper-Bessel and their modified counterparts and the Wright generalized Bessel function with links to their (p, q)-extensions are proposed.
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Jankov Maširević, D., Parmar, R.K. & Pogány, T.K. (p, q)-Extended Bessel and Modified Bessel Functions of the First Kind. Results Math 72, 617–632 (2017). https://doi.org/10.1007/s00025-016-0649-1
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DOI: https://doi.org/10.1007/s00025-016-0649-1
Keywords
- (\(p, q\))-Extended beta function
- (\(p, q\))-Extended Bessel and modified Bessel functions of the first kind
- (\(p, q\))-Extended Struve and modified Struve functions
- Integral representation
- Bounding inequalities
- Complete monotonicity
- Differential–difference equation
- Turán type inequality