the mixture distribution equal to twice the alpha parameter of the mixing distribution—and the beta parameter of the mixing distribution always equal to one). A general closed form expression for stable PDFs with rational values of is available in terms of Meijer G-functions. Fox H-Functions can also be used to express the stable probability density functions. For simple rational numbers, the closed form expression is often in terms of less complicated special functions. Several closed form expressions having rather simple expressions in terms of special functions are available. In the table below, PDFs expressible by elementary functions are indicated by an E and those that are expressible by special functions are indicated by an s. Some of the special cases are known by particular names: For and , the distribution is a Landau distribution (L) which has a specific usage in physics under this name. For and the distribution reduces to a Holtsmark distribution with scale parameter c and shift parameter μ. Also, in the limit as c approaches zero or as α approaches zero the distribution will approach a Dirac delta function . Series representation The stable distribution can be restated as the real part of a simpler integral: Expressing the second exponential as a Taylor series, we have: where . Reversing the order of integration and summation, and carrying out the integration yields: which will be valid for x ≠ μ and will converge for appropriate values of the parameters. (Note that the n = 0 term which yields a delta function in x − μ has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of x − μ which is generally less useful. For one-sided stable distribution, the above series expansion needs to be modified, since and . There is no real part to sum. Instead, the integral of the characteristic function should be carried out on the negative axis, which yields: Simulation of stable variables Simulating sequences of stable random variables is not straightforward, since there are no analytic expressions for the inverse nor the CDF itself. All standard approaches like the rejection or the inversion methods would require tedious computations. A much more elegant and efficient solution was proposed by Chambers, Mallows and Stuck (CMS), who noticed that a certain integral formula yielded the following algorithm: generate a random variable uniformly distributed on and an independent