Expression of the Holtsmark function in terms of hypergeometric _2F_2 and Airy Bi functions

01/30/2020

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by   Jean-Christophe Pain, et al.

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The Holtsmark distribution has applications in plasma physics, for the electric-microfield distribution involved in spectral line shapes for instance, as well as in astrophysics for the distribution of gravitating bodies. It is one of the few examples of a stable distribution for which a closed-form expression of the probability density function is known. However, the latter is not expressible in terms of elementary functions. In the present work, we mention that the Holtsmark probability density function can be expressed in terms of hypergeometric function _2F_2 and of Airy function of the second kind Bi and its derivative. The new formula is simpler than the one proposed by Lee involving _2F_3 and _3F_4 hypergeometric functions.