Home Mathematics Extended Mellin integral representations for the absolute value of the gamma function
Article
Licensed
Unlicensed Requires Authentication

Extended Mellin integral representations for the absolute value of the gamma function

  • Nicolas Privault EMAIL logo
Published/Copyright: January 23, 2018

Abstract

We derive Mellin integral representations in terms of Macdonald functions for the squared absolute value s|Γ(a+is)|2 of the gamma function and its Fourier transform when a<0 is non-integer, generalizing known results in the case a>0. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of a Fokker–Planck equation.

MSC 2010: 32A26; 33C10; 33B15

Award Identifier / Grant number: MOE2016-T2-1-036

Funding statement: This research was supported by the Singapore MOE Tier 2 Grant MOE2016-T2-1-036

References

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Appl. Math. Ser. 55, U. S. Department of Commerce, Washington, 1964. Search in Google Scholar

[2] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge University Press, Cambridge, 1999. 10.1017/CBO9781107325937Search in Google Scholar

[3] D. Chakrabarti and G. K. Srinivasan, On a remarkable formula of Ramanujan, Arch. Math. (Basel) 99 (2012), no. 2, 125–135. 10.1007/s00013-012-0416-9Search in Google Scholar

[4] A. Comtet, C. Monthus and M. Yor, Exponential functionals of Brownian motion and disordered systems, J. Appl. Probab. 35 (1998), no. 2, 255–271. 10.1239/jap/1032192845Search in Google Scholar

[5] M. Craddock, On an integral arising in mathematical finance, Nonlinear Economic Dynamics and Financial Modelling, Springer, Cham (2014), 355–370. 10.1007/978-3-319-07470-2_20Search in Google Scholar

[6] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms. Vol. I, McGraw-Hill, New York, 1954. Search in Google Scholar

[7] S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, 2nd ed., Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Boston, 2002. 10.1007/978-0-8176-8134-0Search in Google Scholar

[8] C. Pintoux and N. Privault, The Dothan pricing model revisited, Math. Finance 21 (2011), no. 2, 355–363. 10.1111/j.1467-9965.2010.00434.xSearch in Google Scholar

[9] S. Ramanujan, Some definite integrals, Messenger 44 (1915), 10–18. Search in Google Scholar

[10] A. Schenzle and H. Brand, Multiplicative stochastic processes in statistical physics, Phys. Rev. A 20 (1979), no. 4, 1628–1647. 10.1103/PhysRevA.20.1628Search in Google Scholar

[11] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1995. Search in Google Scholar

[12] S. B. Yakubovich, A class of polynomials and discrete transformations associated with the Kontorovich–Lebedev operators, Integral Transforms Spec. Funct. 20 (2009), no. 7–8, 551–567. 10.1080/10652460802648473Search in Google Scholar

Received: 2017-9-10
Accepted: 2017-12-29
Published Online: 2018-1-23
Published in Print: 2018-3-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 15.3.2026 from https://www.degruyterbrill.com/document/doi/10.1515/anly-2017-0046/html
Scroll to top button