Document Type : Research Paper


Department of Basic Science and Humanities, NMIMS, MPSTME, Vile-Parle (West), Mumbai-400056, Maharashtra, India.


In this study, the quaternion Hankel transform is developed. Basic operational properties and inversion formula of quaternion Hankel transform are derived. Parseval’s relation for this transform is also established. The generalized quaternion Hankel transform is presented. In the concluding section, we demonstrate the application of the quaternion Hankel transform to Cauchy’s problem.


Main Subjects

[1] R. Agarwal, M.P. Goswami and R.P. Agarwal, Hankel transform in bicomplex space and applications, TJMM, 8 (1) (2016), pp. 1-14.
[2] M. Bahri, R. Ashino and R. Vaillancourt, Continuous quaternion Fourier and wavelet transform, Int. J. Wavelets Multiresolution Inf. Process., 12 (4) (2014), 1460003.
[3] B. Davies, Integral transforms and their Applications, Springer, 1978.
[4] L. S. Dube and J. N. Pandey, On the Hankel transform of distributions, Tohoku Math. J., 27 (3) (1975), pp. 337-354.
[5] A. Elkachkouri, A. Ghanmi and A. Hafoud, Bargmann's versus of the quaternionic fractional Hankel transform, arXiv preprint arXiv:2003.05552, 2020.
[6] I. M. Gelfand and G.E. Shilov, Generalized Functions, Academic Press, New York, 1967.
[7] A. Ghaani Farashahi and G.S. Chirikjian, Fourier-Bessel series of compactly supported convolutions on disks, Anal. Appl., 20 (2) (2022), pp. 171-192.
[8] E. Hitzer, Quaternion Fourier Transform on Quaternion Fields and Generalizations, Adv. Appl. Clifford Algebr., 17 (3) (2007), pp. 497-517.
[9] F.H. Kerr, Fractional powers of Hankel transforms in the Zemanian spaces, J. Math. Anal. Appl., 166(1) (1992), pp. 65-83.
[10] E.L. Koh and C.K. Li, On the inverse of the Hankel transform, Integral Transform Spec. Funct., 2(4) (1994), pp. 279-282.
[11] E.L. Koh and A.H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math., 16 (5) (1968), pp. 945-957.
[12] A.C. Lewis, Chapter 35 - William Rown Hamilton, Lectures on quaternions (1853), Landmark Writings in Western Mathematics, Elsevier Science, 2005.
[13] S.P. Malgonde and L. Debnath, On Hankel type integral transform of generalized functions, Integral Transform Spec. Funct., 15 (5) (2004), pp. 421-430.
[14] S.P. Malgonde and V.R. Lakshmi Gorty, Orthogonal series expansions of generalized functions and the finite generalized Hankel-Clifford transformation of distributions, Rev. Acad. Canar. Cienc., XX (1-2) (2008), pp. 49-61.
[15] J.M. Mendez, The finite Hankel-Schwartz transform, J. Korean Math. Soc., 26 (1) (1989), pp. 43-55.
[16] J.M. Mendez, On the Bessel transformation of arbitrary order, Math. Nachr., 136 (1) (1988), pp. 233-239.
[17] V. Namias, Fractionalization of Hankel transforms, IMA J. Appl. Math., 26 (2) (1980), pp. 187-197.
[18] K. Parmar and V.R. Lakshmi Gorty, One-Dimensional Quaternion Mellin Transform and its applications, Proc. Jangjeon Math. Soc., 24 (1) (2021), pp. 99-112. 
[19] K. Parmar and V.R. Lakshmi Gorty, Application and graphical interpretation of a new two-dimensional quaternion fractional Fourier transform, Int. J. Anal. Appl., 19 (4) (2021), pp. 561-575. 
[20] K. Parmar and V.R. Lakshmi Gorty, Quaternion Stieltjes Transform and Quaternion Laplace-Stieltjes Transform, Commun. Math. Appl., 12 (3) (2021), pp. 633-643.
[21] K. Parmar and V.R. Lakshmi Gorty, Numerical Computation of Finite Quaternion Mellin Transform Using a New Algorithm, International Conference on Advances in Computing and Data Sciences, (2021), pp. 172-182.
[22] E.D. Rainville, Special functions, The Macmillan Company, New York, 1960.
[23] R. Roopkumar, Quaternionic one-dimensional fractional Fourier transform, Optik, 127 (24) (2016), pp. 11657-11661.
[24] I.N. Sneddon, III. Finite Hankel transform, Philos. Mag., 37 (264) (1946), pp. 17-25.
[25] A. Torre, Hankel-type integral transforms and their fractionalization: a note, Integral Transform Spec. Funct., 19 (4) (2008), pp. 277-292.
[26] A.H. Zemanian, Generalized Integral Transformation, John Wiley Sons Inc., New York, 1969.