Abstract
We provide a concrete characterization of the Bergman space of bicomplex-valued bc-meromorphic functions with a strong pole at the origin of the bicomplex discus. The explicit expression of its reproducing kernel is given, and its integral representation as the range of the bicomplex version of the generalized second Bargmann transform is also considered. In addition, we construct the bicomplex analog of the fractional Hankel transform as well as its dual transform. Its range is described and its reproducing kernel is given. Such description involves the zeros of the generalized Laguerre polynomials.
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References
Agarwal, R., Goswami, M.P., Agarwal, R.P.: Hankel transform in bicomplex space and applications. TJMM 8(1), 01–14 (2016)
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C.: Basics of functional analysis with bicomplex scalars, and bicomplex Schur analysis. In: SpringerBriefs in Mathematics. Springer International Publishing, (2014)
Andrews, G.E., Askey, R., Roy, R.: Special functions. Cambridge: Cambridge University Press; (1999). (Encyclopedia of Mathematics and its Applications; vol. 71)
Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, 187–214 (1961)
Bateman, H., Erdélyi, A.: Higher transcendental functions, vol. 2. McGraw-Hill, (1953)
Charak, K.S., Rochon, D.: On factorization of bicomplex meromorphic functions. In: Hypercomplex analysis pp. 55-68 (2008)
Charak, K.S., Rochon, D., Sharma, N.: Normal families of bicomplex meromorphic functions. Ann. Pol. Math. 103(3), 303–317 (2012)
Charak, K.S., Sharma, N.: Bicomplex analogue of Zalcman lemma. Compl. Anal Oper. Theory 8(2), 449–459 (2013)
Elena Luna-Elizarraraz, M., Shapiro, M.E., Struppa, D.C., Vajiac, A.: Bicomplex holomorphic functions: the algebra, geometry and analysis of bicomplex numbers. Birkhäuser, Basel (2015)
Elkachkouri, A., Ghanmi, A., Hafoud, A.: Bargmann’s versus for fractional Fourier transforms and application to the quaternionic fractional Hankel transform. TWMS J. App. Eng. Math. 12(4), 1356–1367 (2022)
Gervais Lavoie, R., Marchildon, L., Rochon, D.: Infinite dimensional bicomplex Hilbert spaces. Ann. Funct. Anal. 1(2), 75–91 (2010)
Ghanmi, A.: On dual transform of fractional Hankel transform. Compl. Var. Elliptic Equ. (2021)
Ghanmi, A., Snoun, S.: Integral representations of Bargmann type for the \(\beta \)-modified Bergman space on punctured unit disc. Bull. Malays. Math. Sci. Soc. 45(3), 1367–1381 (2022)
Ghiloufi, M., Snoun, S.: Zeros of new Bergman kernels. J. Korean Math. Soc. 59(3), 449–468 (2022)
Ghiloufi, M., Zaway, M.: Meromorphic Bergman spaces. Ukr. Math. J. 74(8), 1209–1224 (2023)
Koh, E.L., Zemanian, A.H.: The complex Hankel and I-transformations of generalized functions. J. SIAM Appl. Math. 16(5), 945–957 (1968)
Magnus, W., Oberhettinger, F., Soni, R.P.: Formulas and theorems in the special functions of mathematical physics. Springer-Verlag, Berlin (1966)
Namias, V.: Fractionalization of Hankel transforms. J. Inst. Math. Appl. 26(2), 187–197 (1980)
Namias, V.: The fractional order Fourier transforms and its application to quantum mechanics. J. Inst. Math. Appl 25, 241–265 (1980)
Perez-Regalado, C.O., Quiroga-Barranco, R.: Bicomplex Bergman spaces on bounded domains. ar**v (2018)
Price, G.B.: An introduction to multicomplex spaces and functions. Monographs and Textbooks in pure and applied mathematics, vol. 140. Marcel Dekker Inc., New York (1991)
Rainville, E.D.: Special functions. Chelsea publishing Co., Bronx (1960)
Reséndis, L.E., Tovar, L.M.: Bicomplex Bergman and Bloch spaces. Arab. J. Math. 9, 665–679 (2020)
Ringleb, F.: Beiträge zur funktionentheorie in hyperkomplexen systemen. I. Rendiconti del Circolo Matematico di Palermo 57, 311–340 (1933)
Rochon, D.: On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrödinger equation. Compl. Var. Elliptic Equ. 53(6), 501–521 (2008)
Wiener, N.: Hermitian polynomials and Fourier analysis. J. Math. Phys., 8, pp. 70-73, Collected works Vol. II, 914-918 (1929)
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The author A.Hammam contributed to the entire manuscript. The main ideas that are developed by the author are characterizing the Bergman space of bicomplex-valued bc-meromorphic functions with a strong pole at the origin of the bicomplex discus. He studied the bicomplex extension of the generalized second Bargmann transform and then from that, he constructed the bicomplex fractional Hankel transform as well as its dual transform within some new properties.
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Communicated by Irene Sabadini
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Hammam, A. The Bicomplex Dual Fractional Hankel Transform. Complex Anal. Oper. Theory 18, 30 (2024). https://doi.org/10.1007/s11785-023-01476-z
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DOI: https://doi.org/10.1007/s11785-023-01476-z