The H function (sometimes called Fox’s H function) is a very generally defined special function. It is more general than Meijer’s G-function and is rarely used outside of several references in the literature.
The formal definition (Khan & Pandey, 2017), represented via a Mellin-Barnes type of contour integral is:
Where 0 ≤ m & le; q, 0 ≤ n ≤ p, αj, Βj > 0 and αj, bj are complex numbers, so that poles Γ(bj – Βjs) for j = 1, 2, … m coincide with poles Γ (1 – αj + αjs) for j = 1, 2, … n.
Fox’s H function has some relatively obscure and highly specialized applications, including fractional diffusion (Mainardi, 2005), Mellin transforms and stochastic modeling of wireless communications in a fading environment (Mukasa, 2017). In calculus, it’s occasionally seen in fractional calculus, and is sometimes substituted for the Meijer G function as a better fit for certain pole structures in contours.
Other H Functions
Fox’s H function shouldn’t be confused with an array of “H functions” in computing, including the fast-growing function developed by Chris Bird, the first few values of which can be found here. In R, there is also a (no relation) H function which “calculates the alpha, beta, and gamma ‘standard diversity indices'” (Charney, 2020_.
Al-Musallam, F. A. and Tuan, V. K. “H-Function with Complex Parameters I: Existence.” Int. J. Math. Math. Sci. 25, 571-586, 2001a.
Bird, Chris. Beyond Bird’s Nested Arrays III.
Buschman, R. G. “H-Functions of Two Variables, I.” Indian J. Math. 20, 139-153, 1978.
Fox, C. “The G and H-Functions as Symmetrical Fourier Kernels.” Trans. Amer. Math. Soc. 98, 395-429, 1961.
Charney, N. ‘Standard Diversity Indices’ For Alpha, Beta, And Gamma Diversities. Retrieved August 30, 2020 from: https://www.rdocumentation.org/packages/vegetarian/versions/1.2/topics/H
Khan, A. & Pandey, N. Integrals Involving H-function of Several Complex Variables. International Journal of Scientific and Research Publications, Volume 7, Issue 2, February 2017 95
Mainardi, F. Fox H functions in fractional diffusion. Journal of Computational and Applied Mathematics.Volume 178, Issues 1–2, 1 June 2005, Pages 321-331
Mathai, A. M. and Saxena, R. K. The H-Function with Applications in Statistics and Other Disciplines.0470263806 New Delhi, India: Wiley, 1978.
Mukasa, C. Stochastic Modeling of Wireless Communications in a Fading Environment via Fox’s H-Function. Retrieved August 30, 2020 from: http://fau.digital.flvc.org/islandora/object/fau%3A39777/datastream/OBJ/view/Stochastic_Modeling_of_Wireless_Communications_in_a_Fading_Environment_via_Fox_s_H-Function.pdf
Stephanie Glen. "H Function (Fox’s H-Function)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/h-function-foxs/
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