# (p,q)-Extended Bessel and Modified Bessel Functions of the First Kind

(p,q)-Extended Bessel and Modified Bessel Functions of the First Kind 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 $$J_{\nu ,p,q}$$ J ν , p , q , the (p, q)-extended modified Bessel function $$I_{\nu ,p,q}$$ I ν , p , q of the first kind of order $$\nu$$ ν by making use two additional parameters in the integrand, as well as the (p, q)-extended Struve $$\mathbf{H}_{\nu ,p,q}$$ H ν , p , q and the modified Struve $$\mathbf{L}_{\nu ,p,q}$$ L ν , 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# (p,q)-Extended Bessel and Modified Bessel Functions of the First Kind

, Volume 72 (2) – Jan 11, 2017
16 pages

/lp/springer_journal/p-q-extended-bessel-and-modified-bessel-functions-of-the-first-kind-0hLRtk6PSk
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-016-0649-1
Publisher site
See Article on Publisher Site

### 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 $$J_{\nu ,p,q}$$ J ν , p , q , the (p, q)-extended modified Bessel function $$I_{\nu ,p,q}$$ I ν , p , q of the first kind of order $$\nu$$ ν by making use two additional parameters in the integrand, as well as the (p, q)-extended Struve $$\mathbf{H}_{\nu ,p,q}$$ H ν , p , q and the modified Struve $$\mathbf{L}_{\nu ,p,q}$$ L ν , 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.

### Journal

Results in MathematicsSpringer Journals

Published: Jan 11, 2017

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