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
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.
^Weisstein, Eric W.
"Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03.
^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.
S2CID122652898.
^Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.
^Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey.
arXiv:1004.2950.
Fox, C. (1928). "The asymptotic expansion of integral functions defined by generalized hypergeometric series". Proc. London Math. Soc. 27 (1): 389–400.
doi:
10.1112/plms/s2-27.1.389.
Wright, E. M. (1935). "The asymptotic expansion of the generalized hypergeometric function". J. London Math. Soc. 10 (4): 286–293.
doi:
10.1112/jlms/s1-10.40.286.
Wright, E. M. (1940). "The asymptotic expansion of the generalized hypergeometric function". Proc. London Math. Soc. 46 (2): 389–408.
doi:
10.1112/plms/s2-46.1.389.
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
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.
^Weisstein, Eric W.
"Wright Function". From MathWorld--A Wolfram Web Resource. Retrieved 2022-12-03.
^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.
S2CID122652898.
^Erdelyi, A (1955). The Bateman Project, Volume 3. California Institute of Technology.
^Mainardi, Francesco; Mura, Antonio; Pagnini, Gianni (2010-04-17). The M-Wright function in time-fractional diffusion processes: a tutorial survey.
arXiv:1004.2950.
Fox, C. (1928). "The asymptotic expansion of integral functions defined by generalized hypergeometric series". Proc. London Math. Soc. 27 (1): 389–400.
doi:
10.1112/plms/s2-27.1.389.
Wright, E. M. (1935). "The asymptotic expansion of the generalized hypergeometric function". J. London Math. Soc. 10 (4): 286–293.
doi:
10.1112/jlms/s1-10.40.286.
Wright, E. M. (1940). "The asymptotic expansion of the generalized hypergeometric function". Proc. London Math. Soc. 46 (2): 389–408.
doi:
10.1112/plms/s2-46.1.389.