Generating Function


Also found in: Acronyms, Wikipedia.

generating function

[′jen·ə‚rād·iŋ ‚fəŋk·shən]
(mathematics)
A function g (x, y) corresponding to a family of orthogonal polynomials ƒ0(x), ƒ1(x),…, where a Taylor series expansion of g (x, y) in powers of y will have the polynomial ƒn (x) as the coefficient for the term y n .
A function, g (y), corresponding to a sequence a0, a1, ⋯) where g (y) = a0+ a1 y + a2 y 2+ …. Also known as ordinary generating function.

Generating Function

 

A generating function of the sequence f0, f1,..., fn is the function

assuming that this power series converges for at least one nonzero value of t.

The sequence f0, f1,..., fn,… can be a sequence of numbers or of functions. In the latter case, the generating function depends not only on t but also on the arguments of the functions fn. For example, if fn = aqn, where a and q are constants, the generating function is

If the fn are Fibonacci numbers—that is, if f0 = 0, f1 = 1, fn+2 = fn+1 + fn—we have

If fn = Tn (x) are Chebyshev polynomials—that is, if T0(x) = 1

and Tn (x) = cos (n arc cos x)—then

Knowledge of the generating function of a sequence often makes it easier to study the properties of the sequence. Generating functions are used in probability theory, in the theory of functions, and in the theory of invariants in algebra. Methods involving generating functions were first applied by P. Laplace to solve certain problems in probability theory.

REFERENCES

Feller, W. Vvedenie v teoriiu veroiatnostei i eeprilozheniia, 2nd ed., vols. 1–2. Moscow, 1967. (Translated from English.)
Natanson, I. P. Konstruktivnaia teoriia funktsii. Moscow-Leningrad, 1949.
References in periodicals archive ?
A generating function being convergent both at [rho] = 0 and at large [rho], and having top-bottom symmetry, is finally chosen through the form
The cluster generating function for u is defined as
In Table 1, we give the cumulant generating function, third and fourth cumulants of W for the three distributions mentioned above.
10), we get the explicit expression for marginal moment generating function of lower record values from extended type I generalized logistic distribution can be obtained as
i], and Z is the argument of the generating function.
m] is negative so the function h does not correspond to a generating function of an operad associated to a product of degree 0.
This formula is then used to determine the generating function of [t.
It is now evident that the extension of the above results to the case of the generating function
Probability generating function definitions for the number, weight and chromatographic chain distributions of radicals and polymer must be applied to the corresponding mass balances at each one of the cells in which the studied reactor is divided.
For random p and [lambda], E(x|p,[lambda]) = a p[lambda] is a random variable with moment generating function given by the hypergeometric function, F.
The cumulant generating function is the natural log of the moment generating function: [K.
The net constructed this generating function in a method somewhat analogous to the mathematical technique called Fourier analysis, which enables scientists to approximate any curve by adding together sine waves of different frequency, phase and amplitude Instead of sine waves, however, the net used another trigonometric function, hyperbolic tangents (tanh).