In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) based on ideas of Charles Fox (1928) and E. Maitland Wright (1935):

Upon changing the normalisation

it becomes pFq(z) for A1...p = B1...q = 1.

The Fox–Wright function is a special case of the Fox H-function (Srivastava & Manocha 1984, p. 50):

A special case of Fox–Wright function appears as a part of the normalizing constant of the modified half-normal distribution[1] with the pdf on is given as , where denotes the Fox–Wright Psi function.

Wright function

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The entire function   is often called the Wright function.[2] It is the special case of   of the Fox–Wright function. Its series representation is

 

This function is used extensively in fractional calculus and the stable count distribution. Recall that  . Hence, a non-zero   with zero   is the simplest nontrivial extension of the exponential function in such context.

Three properties were stated in Theorem 1 of Wright (1933)[3] and 18.1(30–32) of Erdelyi, Bateman Project, Vol 3 (1955)[4] (p. 212)

 

Equation (a) is a recurrence formula. (b) and (c) provide two paths to reduce a derivative. And (c) can be derived from (a) and (b).

A special case of (c) is  . Replacing   with  , we have

 

A special case of (a) is  . Replacing   with  , we have  

Two notations,   and  , were used extensively in the literatures:

 

M-Wright function

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  is known as the M-Wright function, entering as a probability density in a relevant class of self-similar stochastic processes, generally referred to as time-fractional diffusion processes.

Its properties were surveyed in Mainardi et al (2010).[5] Through the stable count distribution,   is connected to Lévy's stability index  .

Its asymptotic expansion of   for   is   where    

See also

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References

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  1. ^ a b Sun, Jingchao; Kong, Maiying; Pal, Subhadip (22 June 2021). "The Modified-Half-Normal distribution: Properties and an efficient sampling scheme". Communications in Statistics – Theory and Methods. 52 (5): 1591–1613. doi:10.1080/03610926.2021.1934700. ISSN 0361-0926. S2CID 237919587.
  2. ^ Weisstein, Eric W. "Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03.
  3. ^ Wright, E. (1933). "On the Coefficients of Power Series Having Exponential Singularities". Journal of the London Mathematical Society. Second Series: 71–79. doi:10.1112/JLMS/S1-8.1.71. S2CID 122652898.
  4. ^ Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.
  5. ^ Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey. arXiv:1004.2950.
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