Generating Function

(redirected from Generating functions)

generating function

[′jen·ə‚rād·iŋ ‚fəŋk·shən]
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.


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 ?
Our approach is a non-trivial extension of the methods based on generating functions from deterministic Hamiltonian systems [8, Chapter 4] to SHSs.
In this way, UTP-Hybrid meets the cost-effectiveness needs of smaller exchanges while still offering the liquidity generating functions needed for their future growth in equity and derivative markets.
In this way, UTP-Hybrid meets the cost-effectiveness requirements of smaller exchanges while still offering the liquidity generating functions needed for their future growth in equity and derivative markets.
Clusters, generating functions and asymptotics for consecutive patterns in permutations.
The topics are basic counting methods, generating functions, the pigeonhole principle, Ramsey theory, error-correcting codes, and combinatorial designs.
16) have established recurrence relations for marginal and joint moment generating functions of lgos from power function and generalized exponential distributions.
We have been able to re-assign staff from administrative functions to revenue generating functions, which has significantly reduced the need to bring in agency staff to meet demand at the warehouse during the busy seasonal peaks.
Pathan and Yasmeen, On partly bilateral and partly unilateral generating functions, J.
Reinvestment into these positions is generally an indicator of business health because employers are able to expand outside revenue generating functions.
They begin with a few examples, just to let students get a feel for it, then look at fundamentals of enumeration; the pigeonhole principle and Ramsey's theorem; the principle of inclusion and exclusion; generating functions and recurrence relations; Catalan, Bell, and Stirling numbers; symmetries and the Polya-Redfield method; graph theory; coding theory; Latin squares; balanced incomplete block designs; and linear algebra methods in combinatorics.
The Apostol Bernoulli numbers fin([greater than or equal to]) are defined by means of the generating functions
At last, we determine the generating functions of the dual symplectic rook monoids in Theorem 3.