Wright functions of the second kind and Whittaker functions
In the framework of higher transcendental functions, the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker...
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| Published in | Fractional calculus & applied analysis Vol. 25; no. 3; pp. 858 - 875 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.06.2022
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| Abstract | In the framework of higher transcendental functions, the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker functions in some special cases of fractional order. In addition, we point out two erroneous representations in the literature. |
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| AbstractList | In the framework of higher transcendental functions, the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker functions in some special cases of fractional order. In addition, we point out two erroneous representations in the literature. |
| Author | Consiglio, Armando Paris, Richard B. Mainardi, Francesco |
| Author_xml | – sequence: 1 givenname: Francesco orcidid: 0000-0003-4858-7309 surname: Mainardi fullname: Mainardi, Francesco email: francesco.mainardi@bo.infn.it organization: Department of Physics and Astronomy, University of Bologna, & INFN – sequence: 2 givenname: Richard B. surname: Paris fullname: Paris, Richard B. organization: Division of Computing and Mathematics, University of Abertay – sequence: 3 givenname: Armando surname: Consiglio fullname: Consiglio, Armando organization: Institut für Theoretische Physik und Astrophysik, and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg |
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| Keywords | Wright functions Hypergeometric functions 30B10 26A33 (primary ) 33C20 Whittaker functions 30E15 41A60 Laplace transform Fractional calculus 34E05 |
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| References | Mainardi, F., Tomirotti, M.: On a special function arising in the time fractional diffusion-wave equation. In: Rusev, P., Dimovski, I., Kiryakova, V. (Eds), Transform Methods and Special Functions, 1994 (Proc. Int. Workshop, Sofia 12–17 August 1994), 171–183, Science Culture Technology, Singapore (1995) MainardiFThe fundamental solutions for the fractional diffusion-wave equationApplied Mathematics Letters1996962328141981110.1016/0893-9659(96)00089-4 Mainardi, F., Tomirotti M.: Seismic pulse propagation with constant Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document} and stable probability distributions. Annali di Geofisica 40, 1311–1328 (1997). [E-print http://arxiv.org/abs/1008.1341] ParisRBKochubeiALuchkoYuAsymptotics of the special functions of fractional calculusHandbook of Fractional Calculus with Applications2019BerlinDe Gruyter297325 WhittakerETAn expression of certain known functions as generalised hypergeometric functionsBull. Amer. Math. Soc.1903103125134155805810.1090/S0002-9904-1903-01077-5 Apelblat, A., Mainardi, F.: Applications of the Efros theorem to the function represented by the inverse Laplace transform of s-μexp(-sν)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^{-\mu } \exp (-s^\nu )$$\end{document}. Symmetry 13, Art. 354, 1–15 (2021). https://doi.org/10.3390/sym13020354; E-print arXiv:2012.07068 [math.CA] Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. World Scientific, Singapore (2010) [2nd Ed. in press (2022)] CahoyDOEstimation and simulation for the M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}-Wright functionCommunications in Statistics - Theory and Methods201241814661477290299910.1080/03610926.2010.5432991319.62073 ParisRBVinogradovVAsymptotic and structural properties of the Wright function arising in probability theoryLithuanian Math. J.201656377409353022510.1007/s10986-016-9324-1 Paris, R.B., Consiglio, A., Mainardi, F.: On the asymptotics of Wright functions of the second kind, Fract. Calc. Appl. Anal. 24(1), 54–72 (2021). https://doi.org/10.1515/fca-2021-0003; [E-print arXiv:2103.04284] Mainardi, F.: On the initial value problem for the fractional diffusion-wave equation. In: Rionero, S., Ruggeri (Eds), 7th Conference on Waves and Stability in Continuous Media (WASCOM 1993), pp. 246–251, World Scientific, Singapore (1994) WrightEMThe generalised Bessel function of order greater than oneQuart. J. Math.194011364810.1093/qmath/os-11.1.36 ParisRBKaminskiDAsymptotics and Mellin-Barnes Integrals2001CambridgeCambridge University Press10.1017/CBO9780511546662 Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions. Related Topics and Applications. 2nd Ed., Monographs in Mathematics, Springer Verlag, Berlin (2020). [1st Ed. (2014)] Gorenflo, R., Luchko, Yu., Mainardi, F.: Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2(4), 383–414 (1999). E-print http://arxiv.org/abs/math-ph/0701069 Mainardi, F.: The Linear Diffusion Equation. Lecture Notes in Mathematical Physics, University of Bologna, Department of Physics, 19 pp. (1996–2006) DoetschGIntroduction to the Theory and Application of the Laplace Transformation1974BerlinSpringer10.1007/978-3-642-65690-3 Consiglio, A., Mainardi, F.: Fractional diffusive waves in the Cauchy and signalling problems. In: Beghin, L., Mainardi, F., Garrappa, R. (Eds), Nonlocal and Fractional Operators, SEMA-SIMAI Springer Ser. No 26, pp. 133–153, Springer Nature Switzerland (2021) LipnevichVLuchkoYuThe Wright function: its properties, applications, and numerical evaluationAIP Conference Proceedings20101301614622281011710.1063/1.35266631237.33013 Humbert, P.: Nouvelles correspondances symboliques. Bull. Sci. Math. (Paris, II Ser.) 69, 121–129 (1945) Wright, E.M.: The asymptotic expansion of the generalised Bessel function. Proc. Lond. Math. Soc. (Ser. 2), 38, 286–293 (1934) ErdélyiASwansonCAAsymptotic forms of Whittaker’s confluent hypergeometric functionsMemoirs of the American Mathematical Society1957125150906780127.29602 Mainardi, F., Consiglio, A.: The Wright function of the second kind in mathematical physics. Mathematics 8(6) (SI on Special Functions with Applications in Mathematical Physics), Art. 884, 1–26 (2021). https://doi.org/10.3390/math8060884; [E-print arXiv:2007.02098] LuchkoYuKochubeiALuchkoYuThe Wright functions and its applicationsHandbook of Fractional Calculus with Applications2019BerlinDe Gruyter241268 Olver, F.W., Lozier, D.W. : Boisvert, R.F. and Clark, C.W. (Eds.) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010) ConsiglioALuchkoYuMainardiFSome notes on the Wright functions in probability theoryWSEAS Transactions on Mathematics201918389393 ParisRBExponentially small expansions of the Wright function on the Stokes linesLithuanian Math. J.20145482105318913910.1007/s10986-014-9229-9 ParisRBThe asymptotics of the generalised Bessel functionMath. Aeterna20177381406 Stanković, B.: On the function of E.M. Wright. Publ. Inst. Math. (Beograd, Nouv. Sér.) 10(24), 113–124 (1970) Aceto, L., Durastante, F.: Efficient computation of the Wright function and its applications to fractional diffusion-wave equations. E-print arXiv:2202.00397v2 42_CR8 42_CR9 42_CR13 42_CR15 42_CR16 A Erdélyi (42_CR7) 1957; 1 42_CR17 ET Whittaker (42_CR28) 1903; 10 42_CR18 42_CR19 42_CR1 42_CR2 42_CR5 RB Paris (42_CR21) 2014; 54 EM Wright (42_CR30) 1940; 11 V Lipnevich (42_CR11) 2010; 1301 42_CR10 42_CR25 42_CR27 42_CR29 RB Paris (42_CR22) 2017; 7 Yu Luchko (42_CR12) 2019 A Consiglio (42_CR4) 2019; 18 DO Cahoy (42_CR3) 2012; 41 F Mainardi (42_CR14) 1996; 9 RB Paris (42_CR24) 2001 RB Paris (42_CR23) 2019 G Doetsch (42_CR6) 1974 RB Paris (42_CR26) 2016; 56 42_CR20 |
| References_xml | – reference: Mainardi, F.: The Linear Diffusion Equation. Lecture Notes in Mathematical Physics, University of Bologna, Department of Physics, 19 pp. (1996–2006) – reference: ParisRBKaminskiDAsymptotics and Mellin-Barnes Integrals2001CambridgeCambridge University Press10.1017/CBO9780511546662 – reference: Humbert, P.: Nouvelles correspondances symboliques. Bull. Sci. Math. (Paris, II Ser.) 69, 121–129 (1945) – reference: Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. World Scientific, Singapore (2010) [2nd Ed. in press (2022)] – reference: ParisRBThe asymptotics of the generalised Bessel functionMath. Aeterna20177381406 – reference: Consiglio, A., Mainardi, F.: Fractional diffusive waves in the Cauchy and signalling problems. In: Beghin, L., Mainardi, F., Garrappa, R. (Eds), Nonlocal and Fractional Operators, SEMA-SIMAI Springer Ser. No 26, pp. 133–153, Springer Nature Switzerland (2021) – reference: LipnevichVLuchkoYuThe Wright function: its properties, applications, and numerical evaluationAIP Conference Proceedings20101301614622281011710.1063/1.35266631237.33013 – reference: Wright, E.M.: The asymptotic expansion of the generalised Bessel function. Proc. Lond. Math. Soc. (Ser. 2), 38, 286–293 (1934) – reference: DoetschGIntroduction to the Theory and Application of the Laplace Transformation1974BerlinSpringer10.1007/978-3-642-65690-3 – reference: Mainardi, F.: On the initial value problem for the fractional diffusion-wave equation. In: Rionero, S., Ruggeri (Eds), 7th Conference on Waves and Stability in Continuous Media (WASCOM 1993), pp. 246–251, World Scientific, Singapore (1994) – reference: Mainardi, F., Tomirotti, M.: On a special function arising in the time fractional diffusion-wave equation. In: Rusev, P., Dimovski, I., Kiryakova, V. (Eds), Transform Methods and Special Functions, 1994 (Proc. Int. Workshop, Sofia 12–17 August 1994), 171–183, Science Culture Technology, Singapore (1995) – reference: MainardiFThe fundamental solutions for the fractional diffusion-wave equationApplied Mathematics Letters1996962328141981110.1016/0893-9659(96)00089-4 – reference: CahoyDOEstimation and simulation for the M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document}-Wright functionCommunications in Statistics - Theory and Methods201241814661477290299910.1080/03610926.2010.5432991319.62073 – reference: Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions. Related Topics and Applications. 2nd Ed., Monographs in Mathematics, Springer Verlag, Berlin (2020). [1st Ed. (2014)] – reference: Apelblat, A., Mainardi, F.: Applications of the Efros theorem to the function represented by the inverse Laplace transform of s-μexp(-sν)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^{-\mu } \exp (-s^\nu )$$\end{document}. Symmetry 13, Art. 354, 1–15 (2021). https://doi.org/10.3390/sym13020354; E-print arXiv:2012.07068 [math.CA] – reference: ErdélyiASwansonCAAsymptotic forms of Whittaker’s confluent hypergeometric functionsMemoirs of the American Mathematical Society1957125150906780127.29602 – reference: Stanković, B.: On the function of E.M. Wright. Publ. Inst. Math. (Beograd, Nouv. Sér.) 10(24), 113–124 (1970) – reference: Aceto, L., Durastante, F.: Efficient computation of the Wright function and its applications to fractional diffusion-wave equations. E-print arXiv:2202.00397v2 – reference: Mainardi, F., Consiglio, A.: The Wright function of the second kind in mathematical physics. Mathematics 8(6) (SI on Special Functions with Applications in Mathematical Physics), Art. 884, 1–26 (2021). https://doi.org/10.3390/math8060884; [E-print arXiv:2007.02098] – reference: WrightEMThe generalised Bessel function of order greater than oneQuart. J. Math.194011364810.1093/qmath/os-11.1.36 – reference: WhittakerETAn expression of certain known functions as generalised hypergeometric functionsBull. Amer. Math. Soc.1903103125134155805810.1090/S0002-9904-1903-01077-5 – reference: Paris, R.B., Consiglio, A., Mainardi, F.: On the asymptotics of Wright functions of the second kind, Fract. Calc. Appl. Anal. 24(1), 54–72 (2021). https://doi.org/10.1515/fca-2021-0003; [E-print arXiv:2103.04284] – reference: Mainardi, F., Tomirotti M.: Seismic pulse propagation with constant Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{document} and stable probability distributions. Annali di Geofisica 40, 1311–1328 (1997). [E-print http://arxiv.org/abs/1008.1341] – reference: Olver, F.W., Lozier, D.W. : Boisvert, R.F. and Clark, C.W. (Eds.) NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010) – reference: ParisRBExponentially small expansions of the Wright function on the Stokes linesLithuanian Math. J.20145482105318913910.1007/s10986-014-9229-9 – reference: ParisRBVinogradovVAsymptotic and structural properties of the Wright function arising in probability theoryLithuanian Math. J.201656377409353022510.1007/s10986-016-9324-1 – reference: ConsiglioALuchkoYuMainardiFSome notes on the Wright functions in probability theoryWSEAS Transactions on Mathematics201918389393 – reference: LuchkoYuKochubeiALuchkoYuThe Wright functions and its applicationsHandbook of Fractional Calculus with Applications2019BerlinDe Gruyter241268 – reference: ParisRBKochubeiALuchkoYuAsymptotics of the special functions of fractional calculusHandbook of Fractional Calculus with Applications2019BerlinDe Gruyter297325 – reference: Gorenflo, R., Luchko, Yu., Mainardi, F.: Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2(4), 383–414 (1999). E-print http://arxiv.org/abs/math-ph/0701069 – volume: 41 start-page: 1466 issue: 8 year: 2012 ident: 42_CR3 publication-title: Communications in Statistics - Theory and Methods doi: 10.1080/03610926.2010.543299 – ident: 42_CR25 doi: 10.1515/fca-2021-0003; – start-page: 297 volume-title: Handbook of Fractional Calculus with Applications year: 2019 ident: 42_CR23 – ident: 42_CR27 – ident: 42_CR19 doi: 10.4401/ag-3863 – ident: 42_CR10 – volume: 11 start-page: 36 year: 1940 ident: 42_CR30 publication-title: Quart. J. Math. doi: 10.1093/qmath/os-11.1.36 – ident: 42_CR9 – ident: 42_CR16 – ident: 42_CR5 doi: 10.1007/978-3-030-69236-0_8 – volume: 10 start-page: 125 issue: 3 year: 1903 ident: 42_CR28 publication-title: Bull. Amer. Math. Soc. doi: 10.1090/S0002-9904-1903-01077-5 – volume-title: Asymptotics and Mellin-Barnes Integrals year: 2001 ident: 42_CR24 doi: 10.1017/CBO9780511546662 – volume: 1301 start-page: 614 year: 2010 ident: 42_CR11 publication-title: AIP Conference Proceedings doi: 10.1063/1.3526663 – ident: 42_CR17 doi: 10.3390/math8060884 – volume: 56 start-page: 377 year: 2016 ident: 42_CR26 publication-title: Lithuanian Math. J. doi: 10.1007/s10986-016-9324-1 – volume: 9 start-page: 23 issue: 6 year: 1996 ident: 42_CR14 publication-title: Applied Mathematics Letters doi: 10.1016/0893-9659(96)00089-4 – volume: 7 start-page: 381 year: 2017 ident: 42_CR22 publication-title: Math. 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| Title | Wright functions of the second kind and Whittaker functions |
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